Research Article Automorphism Properties and ...downloads.hindawi.com/journals/amp/2015/584542.pdf · Research Article Automorphism Properties and Classification of Adinkras B.L.Douglas,

Research ArticleAutomorphism Properties and Classification of Adinkras

B L Douglas1 S James Gates Jr2 B L Segler1 and J B Wang1

1School of Physics University of Western Australia Perth WA 6009 Australia2Center for String and Particle Theory Department of Physics University of Maryland College Park MD 20742-4111 USA

Correspondence should be addressed to J B Wang jingbowanguwaeduau

Received 16 March 2015 Accepted 7 July 2015

Academic Editor Pavel Kurasov

Copyright copy 2015 B L Douglas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Adinkras are graphical tools for studying off-shell representations of supersymmetry In this paper we efficiently classify theautomorphism groups of Adinkras relative to a set of local parameters Using this we classify Adinkras according to theirequivalence and isomorphism classes We extend previous results dealing with characterization of Adinkra degeneracy via matrixproducts and present algorithms for calculating the automorphism groups of Adinkras and partitioning Adinkras into theirisomorphism classes

1 Introduction

Several recent studies [1ndash10] have introduced and developeda novel approach to the off-shell problem of supersymmetryIn particular a graph theoretic tool has emerged to tackle thisproblem by encoding representations of supersymmetry intoa family of graphs termed Adinkras This graphical encodinghas the advantage of allowing convenient manipulation ofthese objects with the goal of achieving a deeper understand-ing of the underlying representations

The classification of Adinkras is a natural goal of thiswork and various aspects of this have been dealt withextensively in previous studies [2 4 8 11] in which manyof the properties of Adinkra graphs have been ascertainedThe works of [2 3] relate topological properties of Adinkrasto doubly even codes and Clifford algebras The work in[11] is particularly striking as apparently Betti numbers andfunctions similar to those of catastrophe theory seem on thehorizonHowever wewill not direct our current effort towardthese observations

The GAAC (Garden AlgebraAdinkraCodes) Programbegan [12ndash15] with a series of observations of what appearedto be universal matrix algebra structures that seem to occurin all off-shell supersymmetrical theories Two unexpectedtransformations have occurred since this start First thereemerged Adinkras providing a graphical technology to rep-resent these matrix algebras and second the connection of

Adinkras to codes As the ultimate goal of this program isto provide a definitive classification of all off-shell supersym-metrical theories the appearance of these new unexpecteddiscoveries continues to be encouraging that the goal can bereached despite the pessimism surrounding this over thirty-year-old unsolved problem

The goal of this paper is to present a method of efficientlydistinguishing Adinkras in other words a way of efficientlysolving the graph isomorphism problem restricted to thisparticular class of graphs This is achieved via a classificationof their automorphism group together with an efficient algo-rithm for computing this automorphism group for any givenAdinkra In particular we specify the automorphism groupof an Adinkra in terms of its associated doubly even codeThese codes are characterized in terms of local propertiesof Adinkras such that the codes and hence the associatedequivalence and isomorphism classes of these graphs can beefficiently computed

The structure of the paper is as follows Section 2 providesa formal graph theoretic definition of Adinkras introduc-ing some graph theoretic terms related to their study anddiscussing some basic properties that are derived from thedefinition Section 3 defines the notions of equivalence andisomorphism on the class of Adinkras We relate Adinkragraphs to doubly even codes citing a result from [2] that allAdinkras have an associated code The work of [3] relatingproperties of Adinkras to Clifford algebras is also discussed

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 584542 17 pageshttpdxdoiorg1011552015584542

2 Advances in Mathematical Physics

and we define a standard form for Adinkras used in theproof of later results Section 4 establishes the main resultof this work classifying the automorphism group of valiseAdinkras in terms of the related doubly even code InSection 5 these results are used to generalize some results of[5] and provide a polynomial that partitions Adinkras intotheir equivalence classes In Section 6 this is extended toisomorphism classes of nonvalise Adinkras and we providean associated algorithm that accomplishes this partitioningSection 7 details some of the associated numerical methodsand results and provides some additional examples

2 Adinkra Graphs

21 Graph-Theoretic Notation TheAdinkra graphs dealt within this work are simple undirected bipartite edge-N-partiteand edge- and vertex-colored graphs Note that in previouswork [1ndash3] Adinkras are considered to be directed graphsHowever as this information is naturally encoded into theheight assignment component of the vertex coloring weremove the edge directions here to simplify the analysis

We define a few of the graph theoretic terms below For amore complete treatment see [16]

A simple undirected graph 119866 = (119881 119864) consists of a vertexset 119881 together with an edge set 119864 sube 119881 times119881 of unordered pairsof 119881

A graph is bipartite if its vertex set can be partitioned intotwo disjoint sets such that no edges lie wholly within eitherset Equivalently this is a graph containing no odd-lengthcycles

A graph 119866 = (119881 119864) is edge-N-partite if its edge set 119864can be partitioned into119873 disjoint sets such that every vertexV isin 119881 is incident with exactly one edge from each of these119873 sets In other words a graph is edge-119873-partite if it has a1-factorisation

Finally a coloring of the edge or vertex set of a graphis a partitioning of these sets into different colour classesFormally this restricts the automorphism group of the graphto the subset that setwise stabilises these colour classes thesubset that does not map vertices (resp edges) in one colourclass to vertices (resp edges) in another

22 Definition and Properties Adinkra graphs were intro-duced in [1] to study off-shell representations of super-symmetry Thorough definitions are provided in [2 3 8]although as mentioned above they vary slightly from thedefinition presented here in that for the purposes of this studywe will consider them to be undirected graphs The mostcurrent and complete definition of Adinkras can be found inDefinition 32 in the work of [2] With that in mind we usethe following definition of Adinkras for the remainder of thiswork

Definition 1 An Adinkra graph 119866 = (119881 119864 hgt 120594 120587) is abipartite vertex- and edge-coloured graph with the followingproperties

Each vertex in 119881 is coloured in two ways(i) a colouring 120594 119881 rarr Z2 corresponding naturally to

the bipartition (labelled as bosons and fermions)

(ii) each vertex being given a height assignment hgt119881 rarr

Z such that adjacent vertices are at adjacent heights

The edges are also coloured in two ways

(i) a partition into119873 colour classes corresponding to anedge-119873-partition of the graph

(ii) an edge parity assignment120587 119864 rarr Z2 we term thesetwo edge types dashed and solid

Finally the connections in the graph are essentially binaryin the following manner Every path of length two havingedge colours (119894 119895) defines a unique 4-cycle with edge colours(119894 119895 119894 119895) That is to say of all 4-cycles including these twoedges only one has the edge colours (119894 119895 119894 119895) Such 4-cyclesall have an odd number of dashed edges

We refer to the valence of an Adinkra as its dimensionHence an Adinkra with 119873 edge colours (not counting edgeparity) is an 119873-dimensional Adinkra There will generallybe an assumed ordering of the edge colours from 1 to 119873with the term 119894rsquoth edges or 119894rsquoth edge dimension referring to alledges of the 119894rsquoth colour The edge parity is often referred toas dashedness (see eg [5]) In this work we also refer to itas the switching state of an edge or set of edges motivated bya forthcoming analogy to switching and two graphs Relativeto the edge-colour ordering the switching state of an edge119890 = (119906 V) of colour 119894will be denoted alternately by120587

119894(119906)120587119894(V)

or 120587(119890) (referring to the 119894rsquoth edge of the vertex 119906 or V or simplythe edge 119890)

The above conditions imply several additional propertiesThe edge-119873-partite restriction requires equal numbers ofbosons and fermions hence the bipartition must consistof two sets of equal size Moreover we have the followingimportant property

Lemma 2 The number of vertices of any Adinkra is a powerof two

Proof Since the Adinkra is 119873-regular and edge-119873-partiteand satisfies the 4-cycle property there exists some 119899 isin

Z+ 119899 le 119873 such that for any 119899-length subset of the edgecolours (ie removing all of the edges whose colour doesnot lie within this subset) the Adinkra is simply an 119899-dimensional hypercube or 119899-cube Hence we have that thenumber of vertices is equal to 2119899 Note that if 119899 = 119873then ignoring edge and vertex colouring the resulting graphis simply the 119873-cube Hence we term the correspondingAdinkra the119873-cube Adinkra (in Section 4 we show that thisis unique to119873) Where 119899 = 119873 we denote the correspondingAdinkra to be an (119873 119896) Adinkra where 119896 = 119873 minus 119899

When drawing Adinkra graphs we represent the bipar-tition by black and white vertices As in previous work byDoran et al [3] the height assignments will be representedby arranging the vertices in rows incrementally according toheight

Example 3 The Adinkras of Figure 1 satisfy all requirementslisted above Note that nodes in the same bipartition (bosons

Advances in Mathematical Physics 3

(i) (ii)

Figure 1 (i) A 3-cube Adinkra and (ii) a (4 1) Adinkra

or fermions) must be at the same height of modulo 2 andthat every 4-cycle containing only two edge colours has anodd number of dashedsolid edges In case (i) this is simplyevery 4-cycle however case (ii) also contains 4-cycles withfour edge colours

It is also worthmentioning at this point that the questionsof whether dashed edges correspond to even or odd parityand which of the black or white node sets correspond tobosons or fermions have been left ambiguous as they donot impact on the following analysis and often represent asymmetry in the system Also note that the absolute heightvalues of height assignments are also currently ambiguous asonly relative values of height will be relevant to the followingwork Finally we note that since this work is concerned onlywith questions of equivalence and isomorphism of Adinkrasthe assumption of connectedness will be made throughout tosimplify the analysis without loss of generality

3 Alternative Representations andModels of Adinkras

There are other particularly useful ways of defining andmod-elling Adinkras Before describing these it will be appropriateto introduce some more terminology specifically relating tolinear codes andClifford algebrasWemust also consider howto define notions of equivalence and isomorphism betweenAdinkras

31 Equivalence and Isomorphism Definitions

Definition 4 Given an Adinkra (119881 119864 hgt 120594 120587) the topologyof the Adinkra is defined as (119881 119864) thus giving the underlyingvertex and edge sets with all the colourings (including edgeparity and height assignments) removed When only thecolourings associated with edge parity and height assign-ments are removed the resulting graph (119881 119864 120594) is termed thechromotopology of the original Adinkra

Twooperations relative to a definition of equivalence havebeen defined on Adinkras [8] The vertex loweringraisingoperation consists of changing the height of a given set of ver-tices while preserving the requirement that adjacent verticesare at adjacent heights Hence two Adinkras (119881 119864 hgt1 120594 120587)

and (119881 119864 hgt2 120594 120587) are related by a vertex loweringraisingoperation if and only if forall119906 V isin 119881 |hgt1(119906) minus hgt1(V)| = 1 rArr|hgt2(119906) minus hgt2(V)|

The operation of switching a vertex consists of reversingthe parity of all edges incident with it We note briefly thatthis preserves the property that 4-cycles with only two edgecolours have an odd number of dashed edges

This switching operation is analogous to Seidel switchingof graphs (see [17 18]) in which adjacency and nonadjacencyare swapped In this case adjacency has been replaced byparitydashedness for the purposes of switching Continuingthis analogy we define the switching class of an Adinkra 119866to be the set of all Adinkras that can be obtained from 119866 byswitching some subset of its vertices

Any Adinkras in the same switching class will be consid-ered isomorphic Adinkras related via vertex raisingloweringoperations will be considered equivalent but not necessarilyisomorphic

Hence the definitions of equivalence and isomorphismconsidered here differ slightly Permuting the vertex labelsswitching and raisinglowering operations all preserve equiv-alence whereas the only operations preserving isomorphismare those of switching and permutations of vertex labels

Example 5 All three Adinkras drawn in Figure 2 are inthe same equivalence class however only the first two areisomorphic

An automorphism of an (119873 119896) Adinkra 119866 = (119881 119864 hgt120594 120587) is a permutation of the vertex set 119901 119881 rarr 119881 119901 isin

Sym(2119873minus119896) that is also an isomorphismNote that as the ultimate goal of introducing Adinkras is

a greater understanding of representations of off-shell super-multiplets useful definitions of equivalence and isomor-phism regardingAdinkras should be related to the underlyingsupermultiplets that they encode However notions of iso-morphism applied to supermultiplets are inherently context-specific A pertinent discussion relating to automorphisms ofsupermultiplets can be found in Appendix B of [6] Here wecover a few of the points relating directly to Adinkras In gen-eral supermultiplets can be regarded as a triple (119876R

120601R120595)

where R120601and R

120595are vector spaces corresponding to the

bosonic and fermionic component fields 120601 and 120595 and119876 = 1198761 119876119873 denotes the set of 119873 supercharges or


Figure 2 Three Adinkras belonging to the same equivalence class The first two are isomorphic whereas the third belongs to a separateisomorphism class

supersymmetry transformation operators mapping betweenR120601and R

120595and their derivatives An Adinkra graph 119866 =

(119881 119864 hgt 120594 120587) can then be related to a supermultiplet Mwith each vertex in 119881 representing a component field ofM

Transformations acting on a supermultiplet are groupedinto two classes inner and outer transformations This dis-tinction reflects the property that general supersymmetricmodels will consist of several supermultiplets each consistingof distinct (R

120601R120595) pairs whilst sharing the same set of

supercharges 119876 An inner transformation is one affecting asingle supermultiplet while leaving the others unchangedFor instance permuting the labels of the component fields ofsome (R

120601R120595) pair affects only that supermultiplet However

a transformation of the set 1198761 119876119873 affects all supermul-tiplets of a given supersymmetric model and is hence termedan outer transformation

A transformation induces an automorphism when theresult is indistinguishable from the original when its imageand preimage are the same However ldquoindistinguishabilityrdquois inherently contextual and so there are several notionsof equivalence between supermultiplets Whether an outertransformation induces an automorphism can depend on theparticularmodel being studied hence permutations of the119876-labels may or may not preserve equivalence

Consequently the definitions of equivalence and isomor-phism provided above for Adinkras are not the only possibledefinitions and as such a general classification of Adinkrasmust address a broader class of possible equivalences Onesuch alternative which is considered in Section 61 involvesthe inclusion of edge-colour permutations as an operationpreserving isomorphism As the edge colours (alternativelyknown as edge dimensions) correspond to the superchargesof the underlying supermultiplet this situation simplyinvolves allowing permutations of the set 1198761 119876119873 topreserve isomorphism

To introduce the tools necessary to classify the automor-phism group properties of Adinkras it will first be necessaryto give a brief description of linear codes and Cliffordalgebras

32 Linear Codes A binary linear (119873 119896) code 119862 sube Z1198992consists of a set of 2119896 elements known as codewords whichform a subgroup of (Z1198992 +) Hence for all 119906 V isin 119862 119906 + V isin 119862When dealing with a codeword 119906 of length119873 we will use thenotation119906 = (1199061 1199062 119906119873) where119906119894 is the 119894rsquoth coordinate of119906 When giving explicit examples of codewords we will oftenomit the commas in this representation as all coordinates areelements of Z2 We also consider all subsequent codes to be

both binary and linear these terms are henceforth droppedfrom their description

A generating set of 119862 is a set 119863 sube 119862 consisting of 119896codewords such that all codewords in 119862 can be formed bya linear combination of elements of 119863 The minimal value of119896 for which a generating set exists is called the dimension ofthe code and is invariant with respect to the particular choiceof generating set

The weight of a codeword 119906 denoted by wt(119906) or simply|119906| is the number of 1rsquos in 119906 If every codeword in a code has aweight of 0 (mod 4) the code is called doubly even The innerproduct of two codewords 119906 and V of length119873 is defined as

⟨119906 V⟩ equiv119873

sum

119894=1119906119894V119894 (mod 2) (1)

Then a doubly even code has the property that any two of itscodewords have an inner product of 0

Definition 6 A standard form generating set 119863 of an (119873 119896)code119862 is defined to be of the form119863 = (119868 | 119860) where 119868 is the119896 times 119896 identity matrix and 119860 is a 119896 times (119873 minus 119896)matrix being theremainder of each codeword in119863

Example 7 By applying a permutation of the coordinates ofcodewords and considering an alternative generating set thecode 1198898 generated by (

1 1 1 1 0 0 0 00 0 1 1 1 1 0 00 0 0 0 1 1 1 1

) can be written in termsof a standard form generating set In order to do so first wewill interchange some coordinates bymoving columns1198628 rarr

1198623 rarr 1198622 rarr 1198628 to give ( 1 1 0 1 0 0 1 00 1 0 1 1 1 0 00 0 1 0 1 1 0 1

) We then perform

row operation 1198771 + 1198772 rarr 1198771 to give (1 0 0 0 1 1 1 00 1 0 1 1 1 0 00 0 1 0 1 1 0 1

) which isin the standard form for a generating set

Note that the freedom to interchange coordinates meansthat there are numerous standard form generating sets thatwe could produce which in turn would have different codesassociated with them Given the nature of the standardform however any code with an associated standard formgenerating set will have a unique standard form generatingset

Note also that generating sets will be assumed to beminimal in that the codewords comprising the set arelinearly independentThus a generating set with 119899 codewordscorresponds to a code with 2119899 distinct codewords

Observation 1 For any (119873 119896) code there exists a generatingset 119863 and permutation of the coordinates of the codewords119901 isin Sym(119873) such that119863 is a standard form generating set


(000)

(100)

(110)

(111)

(011)

(010)(001)

(101)

Figure 3 A 3-cube Adinkra shown together with a codeword representation for each vertex

Thenotion of codewords can be effectively used to furnisha further organisation of the edge and vertex sets of anAdinkra If we order the edge colours of an119873-cube Adinkrafrom 1 to 119873 then assign each of these edge colours to thecorresponding coordinate in an119873-length codeword each ofthe 2119873 vertices in the Adinkra can be represented by such acodeword These codewords are allocated in such a way thattwo vertices connected by the 119894rsquoth edge differ precisely in the119894rsquoth element of their respective codewords

Example 8 Applying this codeword representation to thevertices of a 3-cube Adinkra yields the graph of Figure 3Note that for an119873-cube Adinkra two vertices are connectedby an edge if and only if their corresponding codewordrepresentations have a Hamming distance of 1 (differing inonly one position)

An important theorem of [2] links general (119873 119896)Adinkras to doubly even codes We paraphrase it combinedwith other results from the same work below

Theorem9 (see [2]) Every (119873 119896)Adinkra (up to equivalence)class can be formed from the119873-cube Adinkra by the process ofquotienting via some doubly even (119873 119896) code

This quotienting process works as follows We start witha doubly even (119873 119896) code 119862 and an 119873-cube Adinkra witha corresponding edge-colour ordering and codeword repre-sentationWe then identify vertices (and their correspondingedges) related via any codeword in 119862 ensuring first thatidentified vertices have identical edge parities relative to theordering of edge colours In this process an (119873 119896) codeidentifies groups of 2119896 vertices and reduces the order of thevertex set from 2119873 to 2119873minus119896 producing an (119873 119896) AdinkraDoran et al [2] showed that the topology of an Adinkrais uniquely determined by a doubly even code with a 1-1correspondence between the two

Example 10 If we take the 4-cube Adinkra with codewordsgiven by the elements of Z4

2 and we quotient by the doublyeven (4 1) code 119862 = (0000) (1111) we produce the (4 1)Adinkra of Figure 1 Quotienting by 119862 identifies pairs ofcodewords that differ by (1111) (eg (0000) and (1111) (0001)

and (1110)) thus reducing the number of vertices from 16 to8 We observe that this identification is consistent with thetopology of the 4-cube Adinkra in that any edge joining twovertices V1 and V2 of the 4-cube Adinkra can also be identifiedwith an edge joining vertices V1 + (1111) and V2 + (1111) inthe 4-cube Adinkra

33 Clifford Algebras

Definition 11 One will consider the Clifford algebra 119862119897(119899)to be the algebra over Z2 with 119899 multiplicative generators1205741 1205742 120574119899 with the property

120574119894 120574119895 = 120574

119894120574119895+ 120574119895120574119894= 21205751198941198951 (2)

where 120575119894119895

is the Kronecker delta In other words eachof the generators is a root of 1 and any two generatorsanticommute

It will be convenient to view the 4-cycle of Adinkrasin terms of the anticommutativity property of Cliffordgenerators If we consider each position in the codewordrepresentation of a vertex of an 119873-cube Adinkra to corre-spond to a given generator of 119862119897(119873) with the vertex itselfbeing related to the product of these Clifford generatorsthen the 119894rsquoth edge dimension can be naturally viewed as thetransformations corresponding to left Clifford multiplicationby the 119894rsquoth Clifford generator

For example a vertex with codeword representation(01100) corresponds to the product of Clifford generators12057421205743 isin 119862119897(5) and the edge connecting the vertex (01100)to (01101) is related to the mapping 12057421205743 997891rarr (1205745)(12057421205743) =120574212057431205745 Then the condition that every 4-cycle which consistsof exactly two edge colours must have an odd number ofdashed edges corresponds to the anticommutativity propertyof Clifford generators 120574

119894120574119895= minus120574119895120574119894for 119894 = 119895

In connection with this Clifford generator notation wedefine a standard form for the switching state of an 119873-cubeAdinkra

Definition 12 A standard form 119873-cube Adinkra has edgeparity as follows The 119894rsquoth edge of vertex V = (V1 V2 V119873)has edge parity given by


120587119894 (V) equiv (V1 + V2 + sdot sdot sdot + V119894minus1) (mod 2) (3)

This corresponds to left Clifford multiplication of V by 120574119894

Note that this standard form is defined relative to some givencodeword labelling of the vertex set Since fixing the labellingof a single vertex fixes that of all vertices we will sometimesrefer to a standard form as being relative to a source nodebeing the vertex with label (00 sdot sdot sdot 0) or simply standard formrelative to 0

Example 13 The 3-cube Adinkra of Example 8 is in standardform Vertices are ordered into heights relative to theircodewords such that heights range from 0 (at the bottom) to3 and vertices at height 119894 have weight 119894The edges are orderedfrom left to right such that green corresponds to 1205741 and redcorresponds to 1205743

As we have seen many Adinkras can be formed byquotienting the119873-cube Adinkra with respect to some doublyeven code (as we have not yet investigated the case ofGnomonAdinkras defined by the ldquozipperingrdquo process presented in[19] the extension of our results to these cases requiresfurther study) However in practice this method can bequite inefficient The following alternative method for con-structing an (119873 119896) Adinkra with associated code 119862 is dueto 119866 Landweber (personal communication) is used by theAdinkramat software package

(1) Start with the standard form (119873 minus 119896)-cube Adinkrainduced on the first119873 minus 119896 edge dimensions

(2) Find a standard form generating set for119862 of the form119863 = (119868 | 119860)

(3) Associate the (119873 minus 119896 + 119894)th edge dimension with theproduct of Clifford generators given by 119860

119894 the 119894rsquoth

row of 119860(4) The 119894rsquoth edge connects a vertex V to the vertex V sdot 119860

119894

with switching state corresponding to right Cliffordmultiplication of V by 119860

119894 up to a factor of (minus1)119865

The factor of (minus1)119865 termed the fermion number operatorsimply applies a (minus1) factor to fermions and leaves bosonsunchanged This factor is required to ensure the parity ofan edge remains the same in either direction Again notethat the choice of whether even- or odd-weight verticesare bosons remains ambiguous For simplicity we considerodd-weight vertices to be fermions for the purposes of the(minus1)119865 factor with the symmetry encoded by a graph-widefactor of plusmn1 in the switching state of the extra 119896 edgedimensions corresponding to two potentially inequivalent(119873 119896) Adinkras

DefineS120594(119862) to be the set of all Adinkras obtained by the

above construction method using the associated doubly even(119873 119896) code 119862 with fermionboson assignment determinedby 120594 It remains to be seen whether S

120594(119862) coincides with

an equivalence class of (119873 119896) Adinkras with associated code119862 To our knowledge this has not been explicitly provenin previous work hence we present a proof of this asfollows

Theorem 14 Given a doubly even (119873 119896) code 119862 and a fermi-onboson assignment function 120594 for any Adinkra 119866 isin S

120594(119862)

the equivalence class of 119866 coincides with S120594(119862)

Proof As we are only considering equivalence classes theheight assignments will be ignored

Consider an 119873-cube Adinkra quotiented with respect toan (119873 119896) code 119862 with standard form generating set 119863 =

(119868 | 119860) producing an (119873 119896) Adinkra 119866 = (119881 119864 hgt 120594 120587)Assume without loss of generality that the vertices remainingafter quotienting all have fixed (119873 minus 119896 + 1)th rarr 119873thcodeword characters (eg all fixed as 0) Then the rows of 119860represent the vertices identified by the quotienting processWe want to show that up to isomorphism there are atmost two different ways this quotienting operation can beperformed

Isolating any set of 119873 minus 119896 edge dimensions of 119866 yieldsan induced (119873minus119896)-cube Adinkra119867 = (119881 119864

1015840 hgt 120594 120587) with

1198641015840sub 119864 and120587 restricted to1198641015840Wewill see in Section 4 that for

fixed 119898 all 119898-cube Adinkras belong to a single equivalenceclass Hence without loss of generality 119867 can be consideredto be in standard form This in turn fixes the switching stateof all edges belonging to the chosen119873 minus 119896 edge dimensionsNow only a single degree of freedom remains in the switchingstates of the remaining 119896 edge dimensions due to the 4-cycle condition In other words fixing the switching stateof any one of the remaining edges fixes all remaining edgesThis choice corresponds to a graph-wide factor of plusmn1 in theswitching state of the additional 119896 edge dimensions and isdetermined by 120594

Since this graph-wide factor of plusmn1 matches the differencebetween the two possibly inequivalent Adinkras producedby the above construction method it remains to show thatthe construction method does indeed produce valid (119873 119896)Adinkras Hence me must verify that the 4-cycle conditionholds

As odd-weight vertices are considered to be fermions the(minus1)119865 factor can be replaced by an additive factor ofsum119873minus119896

119895=1 119909119895for a given vertex 119909 isin 119881 Then for 119894 gt 119873 minus 119896 119909 isin 119881 119860

119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896

(1198861 + 1198862 + sdot sdot sdot + 119886(119873minus119896minus1)) (mod 2)

(4)

The 4-cycles in 119866 with two edge colours 119894 and 119895 will be splitinto three cases

(i) 119894 119895 le 119873 minus 119896 119894 lt 119895(ii) 119894 le 119873 minus 119896 119895 gt 119873 minus 119896(iii) 119894 119895 gt 119873 minus 119896

Note that a given vertex 119909 isin 119881 defines a unique such 4-cycleIn case (i) consider two antipodal points of such a 4-cycle 119909


(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)

Figure 4 Two (4 1) Adinkras with switching state in standard form

and 119909 + 119894 + 119895 Then the sum of the edge parities of the 4-cycleequals

(1199091 + sdot sdot sdot + 119909119894minus1) + (1199091 + sdot sdot sdot + 119909119895minus1) + (1199091 + sdot sdot sdot + 119909119894minus1)

+ (1199091 + sdot sdot sdot + (119909119895 + 1) + sdot sdot sdot + 119909119895minus1) equiv 1 (mod 2) (5)

Hence the 4-cycle condition holds (note that this is the onlycase present in the construction of the standard form119873-cubeAdinkra)

For case (ii) the antipodal points are 119909 and 119911 where

119911 equiv (1199091 + 1198861 119909119894 + 119886119894 + 1 119909119873minus119896 + 119886119873minus119896)

equiv 119909+119860119895+ 119894 (mod 2)

(6)

and by substituting (4) the sum becomes

(1199091 + sdot sdot sdot + 119909119894minus1) +(119873minus119896

sum

119897=1119909119897+119909 sdot 119860

119895)

+ (1199091 + sdot sdot sdot + 119909119894minus1) + (1198861 + sdot sdot sdot + 119886119894minus1)

+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)

However |119860119895| equiv 3 (mod 4) so |119909| equiv |119911| (mod 2) Then the

first and third terms cancel as do the second and fifth leaving

119909 sdot 119860+ 119911 sdot 119860+ (1198861 + sdot sdot sdot + 119886119894minus1)

equiv 2 (119909 sdot 119860) + 2 (1198861 + sdot sdot sdot + 119886119894minus1) + 1198862 (1198861)

+ 1198863 (1198861 + 1198862) + sdot sdot sdot + 119886119873minus119896 (1198861 + sdot sdot sdot 119886(119873minus119896minus1))

equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +

10038161003816100381610038161003816119860119895minus 210038161003816100381610038161003816 + sdot sdot sdot + 1 equiv

12|119860 minus 1| |119860|

(mod 2)

(8)

Now |119860| equiv 3 (mod 4) hence12|119860 minus 1| |119860| equiv 3 (mod 4)

equiv 1 (mod 2) (9)

and case (ii) is verified

Case (iii) proceeds similarly to case (ii)Hence the graphs produced via this construction method

are indeed (119873 119896) Adinkras Since [2] showed that all (119873 119896)Adinkras can be produced via the quotienting method therecan be at most two equivalence classes of Adinkras with thesame associated code and hence theAdinkras produced fromthe two methods must coincide

Definition 15 A standard form (119873 119896) Adinkra has switchingstate given by the above construction relative to an associateddoubly even code

Example 16 The smallest nontrivial (119873 119896) Adinkra hasparameters119873 = 4 119896 = 1 with associated doubly even code(1111)The standard form ofDefinition 15 is shown in Figure 4for each choice of the graph-wide factor of plusmn1

In this case the two Adinkras formed belong to differentequivalence classes as we demonstrate in the followingsection

This construction provides a much simpler practicalmethod of constructing Adinkras and the standard formwillbe convenient in the proof of later results Note that all (119873 119896)Adinkras have an associated doubly even code withmatchingparameters of length and dimension For these purposes the119873-cubeAdinkra is considered an (119873 0)Adinkrawith a trivialassociated code of dimension 0

4 Automorphism GroupProperties of Adinkras

We now have sufficient tools to derive the main result of thispart classifying the automorphism group of an Adinkra withrespect to the local parameters of the graph namely the asso-ciated doubly even code We start with several observationsregarding the automorphism properties of Adinkras

Lemma 17 Ignoring height assignments the119873-cube Adinkrais unique to119873 up to isomorphism

Note that equivalently wemay consider Adinkras with onlytwo heights to correspond to the assignment of bosons andfermions and we call these valise Adinkras Those with morethan two heights are called nonvalise Adinkras


Proof It suffices to show that any 119873-cube Adinkra can bemapped to standard form via some set of vertex switchingoperations Such a mapping can be trivially constructedFor example start with a single vertex The parity of itsedge set can be mapped to any desired form by a set ofvertex switching operations applied only to its neighboursIn particular the edge parities can be mapped to those ofthe corresponding standard form Adinkra Since the 4-cyclerestriction of Section 22 holds this process can be continuedconsistently for the set of vertices at each subsequent distancefrom this original vertex For example there will be 119873119862

2

vertices at a distance of two from the original vertex andeach of these vertices will belong to one of the 119873119862

2unique

4-cycles defined by the pairs of edges joining the originalvertex to its neighbours For each of these 4-cycles since wehave already fixed two of its edges to have the correct parity(those joining the original vertex to its neighbours) the 4-cycle property (that the cycle must contain an odd numberof dashed edges) ensures that the remaining two edges bothmust either have the correct parity (requiring no vertexswitching) or have incorrect parity (which can be rectifiedby switching the vertex) This process can then be continuedin the same manner for vertices at further distances TheAdinkra resulting from this process will then be in standardform relative to this original vertex regardless of the initialswitching state of the Adinkra

Lemma 18 119873-cube valise Adinkras are vertex-transitive upto the bosonfermion bipartition That is to say given anytwo vertices V1 and V2 of the 119873-cube Adinkra there exists anautomorphism 120574 which maps V1 rarr V2

Proof To show this we begin with a standard form 119873-cubeAdinkra 119866 = (119881 119864 hgt 120594 120587) with switching state defined byleft multiplication by the corresponding Clifford generatorsas in Definition 12 Since 119873-cube Adinkras are unique up toisomorphism in the sense of Lemma 17 this can be donewithout loss of generality

Then the switching state of edge (119909 119910) where 119909 = 119894 sdot 119910 isof the form 1199091 +1199092 + sdot sdot sdot +119909119894minus1 (mod 2) Consider a nontrivialpermutation 120588 119881 997891rarr 119881 Since 120588 is nontrivial there are twovertices 119906 V isin 119881 119906 = V such that 120588 119906 997891rarr V If 120588 is topreserve the topology of 119866 we must have 120588 119909 997891rarr (119906 sdot V)119909forall 119909 isin 119881 Note that 120588 has no fixed points and is completelydefined in terms of (119906 sdot V) Any automorphism 120574 of 119866 whichpermutes 119881 according to 120588 must also switch some set ofvertices in such a way that edge parity is preserved However120588 maps the edge (119909 119910) where 119909 = 119894 sdot 119910 with edge parity1199091 + 1199092 + sdot sdot sdot + 119909119894minus1 (mod 2) to the edge (120588(119909) 120588(119910)) withedge parity of

(1199091 +1199092 + sdot sdot sdot + 119909119894minus1)

+ ((1199061 + V1) + (1199062 + V+ 2) + sdot sdot sdot + (119906119894minus1 + V119894minus1))

(mod 2)

(10)

Hence the change in edge parity of (119909 119910) does not depend oneither endpoint explicitly only on themapping120588 and the edgedimension 119894 In other words 120588 either preserves the switching

state of all edges of a given edge dimension or reverses theparity of all such edges Note that any single edge dimensioncan be switched (while leaving all other edges unchanged) viaa set of vertex switching operations hence an automorphism120574 exists for all such 120588 Furthermore 120574 is unique to 120588 which isin turn unique to the choice of (119906 sdot V)

Note that if 119866 is in standard form we term the automor-phism mapping vertex 119909 to vertex 0 = (00 sdot sdot sdot 0) as putting119866 in standard form relative to 119909 By this terminology 119866 wasinitially in standard form relative to 0

Corollary 19 The 119873-cube Adinkra is minimally vertex-transitive (up to the bipartition) in the sense that the pointwisestabiliser of the automorphism group is the identity and|Aut(119866)| = (12)|119881| = 2119873minus1 In other words no automor-phisms exist that fix any points of the119873-cube Adinkra

Consider any (119873 119896) Adinkra 119866 = (119881 119864 hgt 120594 120587) Thesub-Adinkra 119867 = (119881 119864

1015840 hgt 120594 120587) induced on any set 119878 of

(119873minus119896) edge dimensions of119866will be an (119873minus119896)-cubeAdinkraAlso for all such 119878 there exists some set of vertex switchingoperations such that the induced Adinkra 119867 is in standardform In the case where 119878 = (1 2 119873 minus 119896) the induced119867is in standard form if and only if 119866 is in standard form

Observation 2 Given some (119873 119896) Adinkra containing astandard form (119873 minus 119896)-cube Adinkra induced on the first(119873minus119896) edge dimensions (or equivalently an induced (119873minus119896)-cube Adinkra in any given form) the switching state of the119894rsquoth edge dimension 119894 gt (119873 minus 119896) is fixed up to a graph widefactor of minus1

In particular fixing the parity of the extra edges 119894 where119894 gt (119873 minus 119896) of a single vertex fixes the parity of allextra edges in the graph This is a direct consequence ofthe anticommutativity property of 4-cycles with two edgecolours

This reduces the problem of finding automorphisms ofa general (119873 119896) Adinkra to that of local mappings betweenvertices In particular for any two nodes 119909 119910 isin 119881 therewill be a unique automorphism of the induced (119873 minus 119896)-cube Adinkra mapping 119909 to 119910 This will extend to a fullautomorphism of 119866 if and only if it preserves the switchingstate of the additional 119896 edges of 119909 Hence we arrive at thefollowing result

Theorem 20 Consider an (119873 119896) Adinkra 119866 = (119881 119864 hgt120594 120587) having an associated code 119862 Two vertices 119909 119910 isin

119881 are equivalent (disregarding vertex colouring there is anautomorphism mapping between them) if and only if theirrelative inner products with respect to each codeword in 119862 areequal In other words exist120574 isin Aut(119866) such that 120574 119909 997891rarr 119910 iffforall119888 isin 119862 ⟨119909 119888⟩ equiv ⟨119910 119888⟩ (mod 2)

Proof Consider without loss of generality the case where 119866is in standard form and take 119867 = (119881 119864

1015840 hgt 120594 120587) to be the

sub-Adinkra induced on the first 119873 minus 119896 edge dimensionsBy Lemma 18 119867 is vertex transitive if we disregard vertexcolouring So consider the automorphism 120588 of119867 120588 119909 997891rarr 119910where 119909 119910 isin 119881 Now 120588 consists of two parts a permutation


of the vertex set 120588 V 997891rarr 119909 sdot 119910 sdot V forall V isin 119881 and a set ofvertex switching operations such that the switching state of119867 is preserved

We wish to know when 120588 switches 119909 relative to 119910 (iewhen 119909 is switched but 119910 is not or vice versa and wheneither both or neither are switched) Where 120587

119894(119909) denotes the

switching state of the 119894rsquoth edge of 119909 (in 119866 or119867) we use 1205871015840119894(119909)

to denote the equivalent switching state of 119909 in 120588(119866) or 120588(119867)(we also denote 120588(119866) and 120588(119867) by1198661015840 and1198671015840 resp) Note that119909 in 1198661015840 is the image of 119910 in 119866 under 120588

For edge dimension 119894 le 119873 minus 119896 we see that since 120588 is anautomorphism of119867 120587

119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840

119894(V sdot 119910) forall V isin 119881 For the case 119894 gt 119873 minus 119896 120587

119894(V) is given by (4)

in the proof of Theorem 14 but what about 1205871015840119894(V)

The value of 1205871015840119894(V) can be found by considering the path

119875119894(119909) in119867 joining 119909 to 119909 sdot119860

119894 where A

119894= (1198861 1198862 119886119873minus119896) If

we denote by119860119894the associated product of Clifford generators

119860119894=

119899

prod

119895=1120574119904119895 (11)

where 119904119895denote the 1rsquos of 119860

119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)

Then 120588will preserve the parity of edge (119909 119909 sdot119860119894) if and only if

the number of dashed edges in paths119875119894(119909) and119875

119894(119910) are equal

(mod 2) In other words

120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)

Now 1205871199041(119909) equiv 1199091 + sdot sdot sdot + 1199091199041minus1 (mod 2) 120587

119904119899(119909 sdot 119860

119894) equiv

(11990911198861) + sdot sdot sdot + (119909119904119899119886119904119899) (mod 2) where 119886119894= 1 for 119894 isin 119904

119895 1 le

119895 le 119899 and 119886119894= 0 elsewhere Hence this can be rearranged as

sum

119890isin119875119894(119909)

120587 (119890) equiv 1198862 (1199091) + 1198863 (1199091 +1199092) + sdot sdot sdot

+ 119886119873minus119896

(1199091 + sdot sdot sdot + 119909119873minus119896minus1)

equiv 1199091 (1198862 + sdot sdot sdot + 119886119873minus119896) + sdot sdot sdot

+ 119909119873minus119896minus1 (119886119873minus119896) (mod 2)

(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)

120587(119890)) and note thatsum119890isin119875119894(119909)

120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)

so we have for 119894 gt 119873 minus 119896

120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)

Since |119860119894| equiv 1 (mod 2) we have |119909|(|119860

119894| + 1) equiv 0 (mod 2)

and the first 4 terms cancel leaving

120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)

which completes the proof

Observation 3 Two vertices having the same relative innerproducts with respect to each element of a set 119878 of codewordsalso have the same relative inner product with respect to thegroup generated by 119878 under addition modulo 2

Corollary 21 Conversely to Theorem 20 consider a singleorbit 120601 of the (119873 119896) Adinkra 119866 with associated doubly evencode 119862 If all elements of 120601 have fixed inner product relative toan119873-length codeword 119888 then 119888 isin 119862 the code by which 119866 hasbeen quotiented

Theorem 20 leads to several important corollaries regard-ing the automorphism group properties of Adinkras Recallthat there are at most two equivalence classes of Adinkraswith the same associated doubly even code Theorem 20implies that disregarding the vertex colourings there is infact only one such equivalence class Hence including thevertex bipartition we see that two equivalence classes exist ifand only if the respective vertex sets of bosons and fermionsare setwise nonisomorphic This in turn occurs if and only ifat least one orbit (and hence all orbits) of 119866 is of fixed weightmodulo 2

In [2 5] one such case was investigated for Adinkraswith parameters 119873 = 4 119896 = 1 In this case the Kleinflip operation which exchanges bosons for fermions wasfound in [2 5] to change the equivalence class of the resultingAdinkra We term this property Klein flip degeneracy andnote the following corollary of Theorem 20

Corollary 22 An (119873 119896) Adinkra with associated code 119862has Klein flip degeneracy if and only if the all-1 codeword(11 sdot sdot sdot 1) isin 119862 This in turn occurs only if119873 equiv 0 (mod 4)

Corollary 23 The automorphism group of a valise (119873 119896)Adinkra 119866 = (119881 119864 hgt 120594 120587) has size 2119873minus2119896minus119886 with 2119896+119886 orbitsof equal size where 119886 = 0 if 119862 contains the all-1 codeword and119886 = 1 otherwise

Proof 119866 has 2119873minus119896 nodes By Theorem 20 disregarding thevertex bipartition there are 2119896 orbits of equal size partitioningthe vertex set Including the bipartition these orbits are splitinto 2 once more whenever they contain nodes of variableweight modulo 2 (ie whenever 119862 does not contain the all-1codeword)

5 Characterising Adinkra Degeneracy

In the work of Gates et al [5] the Klein flip degeneracy inthe119873 = 4 case of Example 16 was investigated It was shown


that Adinkras belonging to these two equivalence classes canbe distinguished via the trace of a particular matrix derivedfromeachAdinkraWewill briefly introduce these results andgeneralise them to general (119873 119896) Adinkras

Throughout the following section we will consider an(119873 119896) Adinkra 119866 = (119881 119864 hgt 120594 120587) with correspondingcode 119862 Note that 119866 has 2119873minus119896 vertices each of degree119873

51 Notation

Definition 24 The adjacencymatrix119860(119866) (or simply119860wherethe relevant Adinkra is clear from the context) is a 2119873minus119896 times2119873minus119896 symmetric matrix containing all the information ofthe graphical representation except the vertex colouringassociated with height assignments Each element of themain diagonal represents either a boson or a fermiondenoted by +119894 for bosons and minus119894 for fermions The off-diagonal elements represent edges of119866 numbered accordingto edge dimension with sign denoting dashedness Forexample if position 119860

119909119910= minus4 there is a dashed edge

of the fourth edge colour between bosonfermion 119909 andfermionboson 119910 Hence the sign of the main diagonalelements represents the vertex bipartition the sign of off-diagonal elements represents the edge parity the absolutevalue of off-diagonal elements represents edge colour andthe position of the elements encodes the topology of theAdinkra

Example 25 Consider the 3-cube Adinkra of Example 8If we order the vertex set into bosons and fermions this

Adinkra can be represented by the matrix

119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1

1 minus2 minus3 0 minus119894 0 0 0

2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)

Note that the symmetric nature of the adjacency matrixarises naturally out of the definition and requiring that thisproperty be upheld imposes no further restrictions in and ofitself

We define 119871 and 119877 matrices similarly as in [5] to repre-sent a single edge dimension of the Adinkra 119871 and119877matricesencode this edge dimension with rows corresponding tobosons and fermions respectively As each edge dimensionis assigned a separate matrix the elements corresponding toedges are all set to plusmn1 Then the Adinkra of Example 25 has 119871matrices

1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)

These matrices can be read directly off the top right quadrantof the adjacency matrix Similarly the 119877 matrices can beread off the lower left quadrant In this case we have 119877

119894=

119871119879

119894 where 119879 denotes the matrix transpose Note that 119871 and

119877 matrices taken directly from the adjacency matrix willalways be related viamatrix transpose Furthermore since weare considering only off-shell supermultiplets these matricesmust also be squareWewill assume that all subsequent 119871 and119877matrices are related in this way We define a further object120574i to be a composition of 119871 and 119877matrices of the form

120574119894= [

0 119871119894

1198771198940] (20)

Further details regarding these objects can be found in[5] In particular it was shown in [5] that the Klein flipdegeneracy of several (4 1) Adinkras can be characterised bythe trace of quartic products of these matrices according tothe formula

Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)

where 120575 represents the Kronecker delta and 120598 the Levi-Civitasymbol and where 120594

0= plusmn1 distinguishes between the

(4 1) Adinkras belonging to the two equivalence classes ofExample 16 It is also conjectured in [5] that this samemethodfor distinguishing equivalence classes can be generalised to allvalues of the parameters119873 and 119896 We prove this conjecture inthe following section

52 Degeneracy for General 119873 In order to generalise (21) tohigher 119873 we wish to determine the form taken by the traceof products of these 120574 matrices and establish exactly whenthis can be used to partition Adinkras into their equivalenceclasses Firstly we note the following properties of Adinkras

Lemma 26 The cycles of an 119873-cube Adinkra consist entirelyof paths containing each edge dimension 0 (mod 2) times


Lemma 27 The cycles of an (119873 119896) Adinkra with associatedcode 119862 consist of paths in which

(i) each edge dimension is traversed 0 (mod 2) times(ii) at least one edge dimension is traversed 1 (mod 2)

times

In case (ii) the set of such edge dimensions traversed 1 (mod 2)times corresponds to a codeword in 119862

In fact the codeword (00 sdot sdot sdot 0) is in any such code 119862 soall cycles have this property however case (i) is considered atrivial instance

Consider the product of 119905 120574 matrices of 119866 denoted by119872 = 120574

11989411205741198942sdot sdot sdot 120574119894119905 As we are considering only 119871 and119877matrices

such that 119871119894= 119877119879

119894 all 120574matrices defined as in (20) will be real

and symmetric This leads to the following property for119872

Lemma 28 The elements of the main diagonal of 119872 arenonzero if and only if the path 119875

119872= (1198941 1198942 119894119905) represents

a closed loop within the Adinkra This can occur by one of twoways

(i) trivially if each edge dimension contained in 119901 ispresent 0 (mod 2) times

(ii) if 119901 corresponds to a codeword (or set of codewords) in119862

Proof Consider the case 119875119872= (119894 119895) If 119894 = 119895119872 is trivially

the identity and hence (i) holds If 119894 = 119895 then119872119909119910

= 0 if andonly if vertices 119909 and 119910 are connected via a path (119894 119895) (ie if119909 = 119894 sdot 119895 sdot 119910) Extending to general119872 if119872

119909119909= 0 for some

119909 isin 119881 this implies that 119909 is connected to itself via the path119875119872 a closed loopcycle in119866 Hence Lemma 27 completes the

proof

Corollary 29 119872119909119909

= 0 for some 119909 isin 119881 if and only if this istrue for all 119909 isin 119881

Lemma 30 In case (ii) of Lemma 28 with119872119909119909= plusmn1 for all

119909 isin 119881119872119909119909= 119872119910119910

if and only if 119909 and 119910 have the same innerproduct with respect to the codeword corresponding to 119875

119872

Proof Consider the product of Clifford generators corre-sponding to path 119875

119872 Since (ii) holds after cancelling

repeated elements we are left with some codeword 119901 isin 119862such that 119901 = 119886119875

119872 where 119886 = plusmn1 depending on whether

119875119872corresponds to an even or odd permutation of 119901 relative

to shifting and cancelling of Clifford generators Since 119862 is agroup either

(i) 119901 isin 119863 a standard form generating set of 119862 or(ii) 119901 = (1198921 +1198922 + sdot sdot sdot + 119892119903) where 119892119894 isin 119863 for any such119863

Assume (i) holdsThen 119901 = (1199011 1199012 119901119905) such that 119901119894 le(119873 minus 119896) for 119894 = 119905 and 119901

119905gt (119873 minus 119896) with the first (119873 minus 119896)

edge dimensions defined relative to the particular standardform generating set119863 being considered In other words 119901

119905=

1199011 sdot 1199012 sdot sdot 119901119905minus1 Then forall119909 isin 119866 119909 = (11990911199092 119909119873minus119896) and

the sign of 119872119909119909

omitting a factor of (119886 minus 1)2 is given by(substituting the formulas for 120587

119894(119909) of Section 3)

(1199011 sdot 119909 + 1199012 sdot 119909 + sdot sdot sdot + 119901119905minus1 sdot 119909) + 119909 sdot 119901119905

equiv 1199012 (1199091) + 1199013 (1199091 +1199092) + sdot sdot sdot

+ 119901119905minus1 (1199091 +1199092 + sdot sdot sdot + 119909119905minus2)

+ 1199011 (1199092 + sdot sdot sdot + 119909119905minus1) + sdot sdot sdot + 119901119905minus2 (119909119905minus1) + |119909|

equiv (10038161003816100381610038161199011003816100381610038161003816 minus 1) |119909| minus

119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)

Note that |119901| equiv 0 (mod 4) so the first term disappears Thesame arguments can be followed to show that this holds forcase (ii) alsoThen since the factor of (119886 minus 1)2 is constant forall 119909 isin 119866 it follows that for all 119909 119910 isin 119866 119872

119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩

One immediate corollary of the preceding lemma is thatthe trace of 119872 will vanish whenever 119875

119872corresponds to a

codeword in 119862 In fact Tr(119872) will only be nonzero in thetrivial case where path 119875

119872consists of a set of pairs of edges

of the same colour These are the paths corresponding tothe Kronecker delta terms of (21) In particular this impliesthat Tr(119872) cannot distinguish between equivalence classesof Adinkras directly However if we instead consider powersof 119871 and 119877 matrices the preceding lemma suggests a directgeneralisation of (21) to all (119873 119896) Adinkras

Given an (119873 119896) Adinkra consider the product of 119905 119871matrices 119871

1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we

define value 120590119901such that 120590

119901= 0 if there exists an edge

dimension in 119901 that is present 1 (mod 2) times and 120590119901= 119886

otherwise Here 119886 = plusmn1 corresponding to the sign of therelated product of Clifford generators (since 119901 consists ofpairs of edges of the same colour this product of Cliffordgenerators equals plusmn1) Then we have the following result

Lemma 31 The trace of 1198711198941119871119879

1198942sdot sdot sdot 119871119894119873minus1119871119879

119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)

where 1205940 = plusmn1 depending on the equivalence class of the (119873 119896)valise Adinkras

Proof The 120590119901term follows directly from Lemmas 28 and 30

In Section 4 we show that (119873 119896) Adinkras with the sameassociated code 119862 are all in a single equivalence class exceptwhere (11 sdot sdot sdot 1) isin 119862 In this case the two equivalence classesare related via the Klein flip operation exchanging bosonsand fermions Moreover since 119888 = (11 sdot sdot sdot 1) isin 119862 the Kleinflip operation switches the sign of ⟨119909 119888⟩ for each 119909 isin 119881Then by Lemma 30 the Adinkras from different equivalenceclasses correspond to 1205940 values of opposite sign

Note that if 119873 = 4 and 119901 = (119894 119895 119896 119897) then 120590119901= 120575119894119895120575119896119897minus

120575119894119896120575119895119897+ 120575119894119897120575119895119896 and the result of (21) follows These results can


Figure 5 Two equivalent nonisomorphic Adinkras with the samevalues of 120583(ℎ 119886) for all possible values of ℎ and 119886

be simplified in some respects if we consider the products of 120574matrices instead Recall the definition119872 = 120574

11989411205741198942sdot sdot sdot 120574119894119905 where

119875119872= (1198941 1198942 119894119905) and the fermion number operator (minus1)119865

Instead of taking the trace of 119872 consider the trace of 119872 sdot

(minus1)119865 We have

120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)

and we are considering only 119871 = 119877119879 so for even 119905

119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)

Replacing 119871 matrices by 119877 matrices in Lemma 31 simplychanges the sign of 1205940 so for an (119873 119896) Adinkra

Tr (119872 sdot (minus1)119865) = 2119873minus1198961205940120598119901 (26)

6 Identifying IsomorphismClasses of Adinkras

The results up to this point deal with equivalence classes ofvalise Adinkras where we consider only 2-level AdinkrasFor the nonvalise case we require a method of partitioninggeneral (119873 119896) Adinkras into their isomorphism classes Theessential problem in establishing such amethod is in classify-ing the topology of an Adinkra relative to the automorphismgroup of its underlying 1-level Adinkra (in which vertexcolouring is ignored) Wemight consider simply partitioningthe vertex set into the orbits of this underlying automorphismgroup and then classifying each height by the number of ver-tices of each orbit that it possesses However this method willclearly be insufficient as it ignores the relative connectivitybetween vertices at different heights For example the twoAdinkras of Figure 5 are in the same equivalence class by useof this set of definitions They also have the same numberof vertices from each orbit at each height and yet theyare clearly nonisomorphic no relabelling of the vertices orvertex switching operations canmap between themHowevereven this simple method does come close to partitioningnonvalise Adinkras into their isomorphism classes Recall

that a pointwise stabiliser of the automorphism group ofany Adinkra is the identity no nontrivial automorphismsexist that fix any vertex Note that here the term nontrivialrefers to the corresponding permutation of the vertex setAn automorphism which switches vertices but leaves theordering of the vertex set unchanged is considered trivial Infact fixing any vertex of anAdinkra yields a natural canonicalordering of the vertex set relative to some ordering of theedge dimensions according to the following construction

Construction 1 Suppose we are given an (119873 119896) Adinkra119866 = (119881 119864 hgt 120594 120587) together with an ordering of the edgedimensions of (1198941 1198942 119894119873) Then fix (choose) any vertexV isin 119881 We define an ordering 120582V of the vertex set relative to Vwhere 120582V 119881 997891rarr [2119873minus119896] in the following way

Consider

(i) 120582V(V) = 1(ii) Order the neighbours of V from 2 to119873 + 1 according

to the ordering of the corresponding edge dimensions(120582V(119894119899V) = 1 + 119899 where 1 le 119899 le 119873)

(iii) Repeat this for vertices at distance 2 beginning withthe neighbours of 1198941V and ending with the neighboursof 119894119873V

(iv) Repeat this similarly for vertices at each distance untilall vertices have been assigned an ordering in [2119873minus119896]

In other words 120582V firstly orders the vertex set accordingto distance from V then for each of these sets each vertex 119909 isassigned an ordering based on the lexicographically smallestpath from V to 119909

To formalise the partitioning of heights discussed aboveconsider an (119873 119896) Adinkra 119866 = (119881 119864 hgt 120594 120587) with ℎdifferent height assignments Let Γ

119866be the automorphism

group of the corresponding 1-level Adinkra (ignoring thevertex colouring of 119866) Relative to a particular orderingof the edge dimensions and a particular generating set ofthe associated doubly even codes order the 2119896 orbits of Γ

119866

according to 120591 119881 997891rarr [2119896] We then define 120583119866(ℎ 119886) to be the

number of vertices at height ℎ belonging to the 119886th orbit ofΓ119866 such that

120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)

Consider a pair of Adinkras 119866 = (119881 119864 hgt1 1205941 1205871) and119867 =

(119881 1198641015840 hgt2 1205942 1205872) belonging to the same equivalence class

both having 119905 distinct heights Then 119866 and 119867 have the sameassociated code 119862 If 120583

119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905

relative to a given edge-colour ordering and generating setof 119862 then choose any vertex V isin 119881 which when viewed ineach Adinkra (as the two Adinkras share the same vertex set)belongs to the same orbit and height Relative to this vertex Vconsider the unordered set

cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)

Theorem 32 Consider two Adinkras 119866 = (119881 119864 hgt1 1205941 1205871)and 119867 = (119881 119864

1015840 hgt2 1205942 1205872) and a vertex V isin 119881 with the


Figure 6 Two equivalent nonisomorphic (5 1) Adinkras Theydiffer precisely in the switching state of the yellow edges

properties described above Then cert119866(V) = cert

119867(V) if and

only if 119866 and119867 are isomorphic

Proof If 119866 and 119867 are isomorphic then this is trivially trueConversely assume cert

119866(V) = cert

119867(V) Then since 119866 and119867

belong to the same equivalence class and 120591119866(V) = 120591

119867(V) there

exists an isomorphism 120574mapping119866 to119867 (ignoring the vertexcolourings)Then 120574 extends to a full isomorphism (includingthe vertex colourings) if it preserves height assignmentsThisfollows directly from cert

119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic

The preceding theorem provides an efficient method ofclassifying Adinkras according to their isomorphism classNote that the classification is relative to a given ordering ofthe edge colours and requires a knowledge of the associateddoubly even code In cases where the associated code isunknown Lemma 27 suggests an efficientmethod for findingthe code and hence relating a given (119873 119896) Adinkra to itsldquoparentrdquo 119873-cube Adinkra in the sense of Theorem 9 Inparticular Lemma 27 implies the following result

Corollary 33 Given an (119873 119896) Adinkra with related code 119862the codewords of 119862 correspond exactly to the cycles of 119866 inwhich at least 1 edge dimension appears once (modulo 2)

We provide an example of the above certificates belowConsider the two Adinkras of Figure 6These are both height3 (5 1) Adinkras They are in the same equivalence classhowever by calculating the above certificate for each we showthat they are nonisomorphic

To analyse the above Adinkras we first order the edgecolours from green to purple such that in the codeword

representation green corresponds to the last coordinate andpurple to the first Note that henceforth we will order theright-most coordinate to the left-most coordinate consis-tent with a bit-string interpretation of codewords in thecomputational application Alternatively the green edges areassociated with the first Clifford generator and the purpleedges with the fifth generator As these are (5 1) Adinkrasthey have an associated (5 1) doubly even code By inspectionor by explicitly finding the nontrivial cycles in the Adinkraswe note that the associated code is (11110) corresponding toa 4-cycle comprising edge colours red yellow blue purpleBy Theorem 20 the induced 1-level Adinkra has two orbitscorresponding to the sets of nodeswith the same parity (innerproduct modulo 2) relative to this code Denote the twoAdinkras by 119866 and 119867 respectively Then by applying theresults of Section 4 we obtain 120583 values of

120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)

In other words the two height 3 vertices of119866 are in a differentorbit to those of 119867 Hence cert

119866= cert

119867 and the two

Adinkras are in different isomorphism classes

61 Partitioning into Isomorphism Classes Further ExamplesIn [20] it was shown that simply recording the numberof vertices at each height is not sufficient to characteriseAdinkras In particular they provide examples of pairs ofAdinkras which are in the same equivalence class but notisomorphic despite having the same number of vertices ateach height In this section we analyse the examples of [20]using the techniques described above

Firstly however we note that a different definition ofisomorphism is considered in [20] Specifically they con-sider the situation where permutations of the edge colourspreserve isomorphism a variation discussed in Section 31Allowing this more general definition of isomorphism resultsin several changes to the results of the preceding sectionsIn particular note that the automorphism group of the 119873-cube Adinkra would simply become that of the 119873-cube thehyperoctahedral group of order 2119873119873 A permutation of theedge colours corresponds to a reordering of the codewordassociated with each vertex (or equivalently a permutation ofthe Clifford generators associatedwith each edge dimension)


Hence the automorphism group of (119873 119896) Adinkras will beextended according to the following lemma

Lemma 34 Allowing permutations of the edge colours topreserve isomorphism an (119873 119896) Adinkra 119866 with associateddoubly even code 119862 has an automorphism group (ignoringheight assignments) of order

2119873

22119896+a|Aut (119862)| (30)

where Aut(119862) is the automorphism group of the code 119862 andwhere 119886 = 1 if 119862 contains the all-1 codeword and 119886 = 0otherwise

In otherwords any permutations that correspond to sym-metries of the code will extend naturally to automorphismsof the Adinkra Conversely if a permutation of the codewordis not in the automorphism group of the code then triviallyit cannot be an automorphism of the Adinkra Note thatLemma 34 applies trivially to 119873-cube Adinkras for which119896 = 0 and 119862 = (00 sdot sdot sdot 0) hence |Aut(119862)| = 119873

Hence Klein flip degeneracy no longer necessarilyexhibits for Adinkras containing the all-1 codeword Insteadnote that any automorphism of the associated doubly evencode corresponding to an odd permutation of the edgecolours (or alternatively the underlying supercharges) effec-tively performs a Klein flip substituting bosons for fermionsand vice versa Hence Klein flip degeneracy occurs if andonly if the associated code contains the all-1 codewordand the automorphism group of the code contains no oddpermutations hence narrowing down the possible candidates

Somewhat fortuitously a recent result regarding doublyeven codes sheds some light on this situation Doran et al[20] prove that the automorphism group of self-dual doublyeven codes is always contained in the alternating groupGivena doubly even binary (119873 119896) code 119862 its dual 119862perp is given by

119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)

A self-dual code is simply a code equal to its own dual adoubly even (119873 119896) code is self-dual if and only if 119896 = 1198732and119873 equiv 0(mod 8) Hence the following result holds

Lemma 35 An (119873 119896) Adinkra exhibits Klein flip degeneracy(relative to this definition of isomorphism allowing edge-colourpermutations) if119873 = 2119896 and119873 equiv 0(mod 8)

However the question of whether any other Adinkraswith Klein flip degeneracy (again allowing for edge-colourpermutations) exist still remains This is equivalent to askingwhether any doubly even codes exist which are both not self-dual and contain no nontrivial even-permutation automor-phisms and is the subject of continuing work Furthermoreif the automorphism group of doubly even codes turns out tobe sufficiently rich a greatly simplified certificate encodingthe isomorphism class of an Adinkra could be developed aswill be discussed briefly in Section 8

In [20] two pairs of equivalent but nonisomorphicAdinkras are presented each pair having the same number

Figure 7 Two nonisomorphic height 3 (5 1) Adinkras belongingto the same equivalence class

of vertices at each height The first pair comprises two height3 (5 1) Adinkras isomorphic (up to a permutation of edgecolours) to those of Figure 7 Note that the second Adinkraof Figure 7 is identical to the first Adinkra of Figure 6

Applying the methods described in the previous sectionto this pair proceeds as in the analysis of the Adinkras ofFigure 6 As in that example after ordering the edge coloursfrom blue to green each Adinkra has associated code (11110)We see that the top two nodes of the first Adinkra belong todifferent orbits of the corresponding 1-level Adinkra whereasthe top two nodes of the second Adinkra belong to the sameorbit Hence the two Adinkras are trivially distinguishedIn particular following the conventions in the equations ofExample (29) the 120583 values of these two Adinkras labelled by119866 and119867 respectively are

120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866

= cert119867 and the two Adinkras are in different

isomorphism classes Also note that the ldquoorbit spreadrdquothe number of vertices in each orbit at each height hasa different character in each Adinkra So regardless of theordering of the orbits these two Adinkras must remain indifferent isomorphism classes if edge-colour permutationsare allowed

The second pair of Adinkras in [20] comprises twoheight 3 (6 2) Adinkras isomorphic to the pair displayed inFigure 8

In this case after ordering the edge colours fromdark blueto green as for the top left node of the first Adinkra bothAdinkras have the associated code generated by ( 1 1 1 1 0 0

0 0 1 1 1 1 )and hence each has four orbits in the automorphism group ofthe corresponding 1-level Adinkras The top two nodes of thefirst Adinkra are connected via the length 2 paths (001001)and (000110) having odd inner product with each of thecodewords in the generating set above Hence these nodesare in different orbits of the automorphismgroupConverselythe top twonodes of the secondAdinkra are connected via thelength 2 paths (110000) (001100) and (000011) having eveninner product with each of the codewords Hence they arein the same orbit As a result the certificates of Theorem 32are different for each Adinkra and hence they are in differentisomorphism classes Again following the conventions in theequations of Example (29) we obtain 120583 values of

120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)

Hence as in the previous example cert119866

= cert119867 and

moreover the Adinkras remain in different isomorphismclasses if edge-colour permutations are allowed

7 Numerical Results

The automorphism group results of the preceding sectionswere also verified numerically independently to the analyt-ical results All (119873 119896)Adinkras up to119873 le 16 were producedand the related automorphism group and equivalence classeswere calculated for each suchAdinkraThe results foundwereconsistent with the analytical results described in the earliersections In particular

(i) all (119873 119896) Adinkras with the same associated code 119862were found to be in the same equivalence class exceptin the cases where (11 sdot sdot sdot 1) isin 119862 In these casesthe Adinkras were split into two equivalence classes


related via the Klein flip operation as described inCorollary 22

(ii) all 1-level (119873 119896) Adinkras had 2119896 orbits each con-sisting of sets of vertices with the same set of innerproducts with the codewords in 119862 according toTheorem 20

All doubly even (119873 119896) codes up to 119873 = 28 (and manylarger parameter sets) can be found online (Miller [21])For each parameter set (119873 119896) the two possibly inequivalentstandard form Adinkras corresponding to the constructionmethod ofTheorem 14 were produced relative to each doublyeven code on these parameters As we were consideringonly equivalence classes and automorphism groups of valiseAdinkras it sufficed to use the adjacency matrix formdefined in Section 5 for this analysis as height assignmentsneed not be encoded in the representation The orbits werethen calculated from the adjacency matrices by forming acanonical form relative to each node via a set of switchingoperations and a permutation of the vertex set A canonicalform is defined to be amapping120587 consisting of a permutationof the vertex labels and set of vertex switching operationssuch that for any two Adinkras 119866 and 119867 120587(119866) = 120587(119867) ifand only if 119866 and119867 are isomorphic

Then given a vertex V and adjacencymatrix119860 of an (119873 119896)Adinkra (together with an ordering of the edge colours) wedefine the following canonical form of 119860 relative to V

(i) Permute the ordering of the vertices relative to theirconnections to V and the ordering of edge colourssuch that the vertices are ordered

(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1

sdot sdot sdot 1198941V) (35)


Figure 9 Standard form valise Adinkra

Figure 10 Adinkra after step (i)

(ii) Switch vertices (1198941V 119894119873V) such that all edges of Vhave even parity

(iii) Repeat for neighbours of vertex 1198941V(iv) Repeat for each vertex in the ordering above such

that edges appearing lexicographically earlier in theadjacency matrix are of even parity where possible

This defines a unique switching state of the Adinkra up toisomorphism Furthermore it is a canonical form in that anytwo vertices belonging to the same orbit result in identicalmatrix forms To illustrate the above process we consider a(6 2) Adinkra with associated code generated by ( 1 0 1 1 1 0

0 1 1 1 0 1 )The standard form valise Adinkra is shown in Figure 9 wherethe top leftmost vertex has codeword (0000) and its edges areordered lexicographically from left to right

Consider the process of converting this Adinkra tocanonical form relative to vertex (0011) (the fifthwhite vertexfrom the left in Figure 9) If we order the vertices by heightfirst then from left to right step (i) corresponds to the set ofvertex lowering operations leading to the Adinkra shown inFigure 10

Steps (ii) and (iii) correspond to a set of switchingoperations permuting the switching state in the followingstages as shown in Figure 11

Step (iv) then involves the remaining set of transforma-tions shown in Figure 12

At this point the Adinkra is now in canonical form Theparticular switching state of this resulting Adinkra is uniqueto choices of source vertices in the same orbit in this case theset of vertices (0000) (0011) (1110) (1101)

8 Conclusions and Future Work

This work provides a graph theoretic characterisation ofAdinkras in particular classifying their automorphismgroups according to an efficiently computable set of localparameters In the current work a number of compar-isons are made to previous work The connection betweenAdinkras and codes [4] has been reexamined and found

to be robust However this connection is also exploited toutilise the standard form leading to more computationallyefficient algorithms for the study of Adinkras As well theobservations based on matrix methods used within thecontext of 119889 = 119873 = 4 [5] have now been extended by aformal proof to all values of 119889 and119873 Also as emphasised inSection 7 numerical studies up to values of 119873 = 16 provideadditional concurrence These results support the proposalthat 1205940 ldquochi-nullrdquo is a class valued function defined on valiseAdinkras

All nonvalise Adinkras through a series of node raisingand lowering can be brought to the form of a valise AdinkraIn this sense 1205940 is defined for all Adinkras However fornonvalise Adinkras 1205940 is not sufficient to define classes Forthis purpose the new certificate 120583A(ℎ 119886) where A is anarbitrary Adinkra seems to fill in a missing gap

Additionally the classificationmethod used here is robustin the sense that it can be readily adapted to address a varietyof definitions of isomorphismequivalence relevant to theparticular supersymmetric system of interest For instance inSection 61 the generalised definition of isomorphism includ-ing edge-colour permutations yields a generalised certificate

81 Open Questions It is the work of future investigations toexplore whether these tools (1205940 and 120583A(ℎ 119886)) are sufficientto attack the problem of the complete classification of one-dimensional off-shell supersymmetrical systems One obvi-ous future avenue of study is to investigate the role codes playin Gnomon Adinkras [19] and Escheric Adinkras [1] Thisas part of continuing to attack the general problem presentscontinuing challenges

Note that when edge-colour permutations are allowedthe calculation of the certificate encoding the isomorphismclass of an Adinkra as detailed in Section 6 is currently morecomplex compared to the situation when edge-colour per-mutations do not preserve isomorphism in that the automor-phism group of the doubly even codemust also be calculatedOne future direction of research is the investigation of theproperties of these automorphism groups It is already knownthat self-dual doubly even codes have automorphism groupscontained in the alternating group [22] however furtherproperties must be discovered to determine all possible Kleinflip degenerate Adinkras in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Akihiro Munemasa fordetailed comments and suggestions on this paper as well asCheryl Praeger Cai-Heng Li Gordon Royle and John Bam-berg for several useful discussions This work was partiallysupported by the National Science Foundation Grant PHY-13515155The authors would also like to thank the Institute ofAdvanced Studies at The University of Western Australia for


rArr

Figure 11 Adinkra after steps (ii) and (iii)

rArr

Figure 12 Adinkra after step (iv)

providing travel support which enabled initial discussionsamong the authors

References

[1] M Faux and S J Gates Jr ldquoAdinkras a graphical technology forsupersymmetric representation theoryrdquo Physical Review D vol71 Article ID 065002 2005

[2] C F Doran M G Faux S J Gates Jr et al ldquoTopology types ofadinkras and the corresponding representations of N-extendedsupersymmetryrdquo httparxivorgabs08060050

[3] C F Doran M G Faux S J Gates Jr et al ldquoAdinkras for clif-ford algebras and worldline supermultipletsrdquo httparxivorgabs08113410

[4] C F Doran M G Faux S J Gates Jr et al ldquoRelating doubly-even error-correcting codes graphs and irreducible repre-sentations of N-extended supersymmetryrdquo httparxivorgabs08060051

[5] S J Gates Jr J Gonzales B MacGregor et al ldquo4D N = 1supersymmetry genomics (I)rdquo Journal of High Energy Physicsvol 2009 no 12 article 008 2009

[6] M G Faux S J Gates Jr and T Hubsch ldquoEffective symmetriesof the minimal supermultiplet of the N = 8 extended worldlinesupersymmetryrdquo Journal of Physics A Mathematical and Theo-retical vol 42 no 41 Article ID 415206 2009

[7] M G Faux K M Iga and G D Landweber ldquoDimensionalenhancement via supersymmetryrdquo Advances in MathematicalPhysics vol 2011 Article ID 259089 45 pages 2011

[8] C F Doran M G Faux S J Gates Jr T Hubsch K MIga and G D Landweber ldquoOn graph-theoretic identificationsof adinkras supersymmetry representations and superfieldsrdquoInternational Journal ofModernPhysics A vol 22 no 5 pp 869ndash930 2007

[9] K Burghardt and S J Gates Jr ldquoAdinkra isomorphisms andlsquoseeingrsquo shapes with eigenvaluesrdquo httparxivorgabs12122731

[10] K Burghardt and S J Gates Jr ldquoA computer algorithm forengineering off-shell multiplets with four supercharges on theworld sheetrdquo httparxivorgabs12095020

[11] S Naples Classification of Adinkra graphs httpmathbardedustudentpdfssylvia-naplespdf

[12] S J Gates Jr and L Rana ldquoA theory of spinning particles forlarge N-extended supersymmetryrdquo Physics Letters B vol 352no 1-2 pp 50ndash58 1995

[13] S J Gates Jr and L Rana ldquoA theory of spinning particles forlarge N-extended supersymmetry (II)rdquo Physics Letters B vol369 no 3-4 pp 262ndash268 1996

[14] S J Gates Jr W D Linch J Phillips and L Rana ldquoThefundamental supersymmetry challenge remainsrdquo Gravitationand Cosmology vol 8 no 1 pp 96ndash100 2002

[15] S J Gates Jr W D Linch and J Phillips ldquoWhen superspace isnot enoughrdquo httparxivorgabshep-th0211034

[16] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[17] A Brouwer and W Haemers Spectra of Graphs Springer 2012[18] J van Lint and J Seidel ldquoEquilateral point sets in elliptic geome-

tryrdquo Indagationes Mathematicae vol 28 pp 335ndash348 1966[19] C F Doran M G Faux S J Gates Jr T Hubsch K M Iga and

G D Landweber ldquoAdinkras and the dynamics of superspaceprepotentialsrdquo httparxivorgabshep-th0605269

[20] C F Doran M G Faux S J Gates Jr T Hubsch K MIga and G D Landweber ldquoA counter-example to a putativeclassification of 1-dimensional N-extended supermultipletsrdquoAdvanced Studies in Theoretical Physics vol 2 article 99 2008

[21] R L Miller ldquoDoubly even codesrdquo httpwwwrlmillerorgdecodes

[22] A Gunther and G Nebe ldquoAutomorphisms of doubly even self-dual binary codesrdquo Bulletin of the LondonMathematical Societyvol 41 no 5 pp 769ndash778 2009

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Algebra

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Stochastic AnalysisInternational Journal of


and we define a standard form for Adinkras used in theproof of later results Section 4 establishes the main resultof this work classifying the automorphism group of valiseAdinkras in terms of the related doubly even code InSection 5 these results are used to generalize some results of[5] and provide a polynomial that partitions Adinkras intotheir equivalence classes In Section 6 this is extended toisomorphism classes of nonvalise Adinkras and we providean associated algorithm that accomplishes this partitioningSection 7 details some of the associated numerical methodsand results and provides some additional examples

2 Adinkra Graphs

21 Graph-Theoretic Notation TheAdinkra graphs dealt within this work are simple undirected bipartite edge-N-partiteand edge- and vertex-colored graphs Note that in previouswork [1ndash3] Adinkras are considered to be directed graphsHowever as this information is naturally encoded into theheight assignment component of the vertex coloring weremove the edge directions here to simplify the analysis

We define a few of the graph theoretic terms below For amore complete treatment see [16]

A simple undirected graph 119866 = (119881 119864) consists of a vertexset 119881 together with an edge set 119864 sube 119881 times119881 of unordered pairsof 119881

A graph is bipartite if its vertex set can be partitioned intotwo disjoint sets such that no edges lie wholly within eitherset Equivalently this is a graph containing no odd-lengthcycles

A graph 119866 = (119881 119864) is edge-N-partite if its edge set 119864can be partitioned into119873 disjoint sets such that every vertexV isin 119881 is incident with exactly one edge from each of these119873 sets In other words a graph is edge-119873-partite if it has a1-factorisation

Finally a coloring of the edge or vertex set of a graphis a partitioning of these sets into different colour classesFormally this restricts the automorphism group of the graphto the subset that setwise stabilises these colour classes thesubset that does not map vertices (resp edges) in one colourclass to vertices (resp edges) in another

22 Definition and Properties Adinkra graphs were intro-duced in [1] to study off-shell representations of super-symmetry Thorough definitions are provided in [2 3 8]although as mentioned above they vary slightly from thedefinition presented here in that for the purposes of this studywe will consider them to be undirected graphs The mostcurrent and complete definition of Adinkras can be found inDefinition 32 in the work of [2] With that in mind we usethe following definition of Adinkras for the remainder of thiswork

Definition 1 An Adinkra graph 119866 = (119881 119864 hgt 120594 120587) is abipartite vertex- and edge-coloured graph with the followingproperties

Each vertex in 119881 is coloured in two ways(i) a colouring 120594 119881 rarr Z2 corresponding naturally to

the bipartition (labelled as bosons and fermions)

(ii) each vertex being given a height assignment hgt119881 rarr

Z such that adjacent vertices are at adjacent heights

The edges are also coloured in two ways

(i) a partition into119873 colour classes corresponding to anedge-119873-partition of the graph

(ii) an edge parity assignment120587 119864 rarr Z2 we term thesetwo edge types dashed and solid

Finally the connections in the graph are essentially binaryin the following manner Every path of length two havingedge colours (119894 119895) defines a unique 4-cycle with edge colours(119894 119895 119894 119895) That is to say of all 4-cycles including these twoedges only one has the edge colours (119894 119895 119894 119895) Such 4-cyclesall have an odd number of dashed edges

We refer to the valence of an Adinkra as its dimensionHence an Adinkra with 119873 edge colours (not counting edgeparity) is an 119873-dimensional Adinkra There will generallybe an assumed ordering of the edge colours from 1 to 119873with the term 119894rsquoth edges or 119894rsquoth edge dimension referring to alledges of the 119894rsquoth colour The edge parity is often referred toas dashedness (see eg [5]) In this work we also refer to itas the switching state of an edge or set of edges motivated bya forthcoming analogy to switching and two graphs Relativeto the edge-colour ordering the switching state of an edge119890 = (119906 V) of colour 119894will be denoted alternately by120587

119894(119906)120587119894(V)

or 120587(119890) (referring to the 119894rsquoth edge of the vertex 119906 or V or simplythe edge 119890)

The above conditions imply several additional propertiesThe edge-119873-partite restriction requires equal numbers ofbosons and fermions hence the bipartition must consistof two sets of equal size Moreover we have the followingimportant property

Lemma 2 The number of vertices of any Adinkra is a powerof two

Proof Since the Adinkra is 119873-regular and edge-119873-partiteand satisfies the 4-cycle property there exists some 119899 isin

Z+ 119899 le 119873 such that for any 119899-length subset of the edgecolours (ie removing all of the edges whose colour doesnot lie within this subset) the Adinkra is simply an 119899-dimensional hypercube or 119899-cube Hence we have that thenumber of vertices is equal to 2119899 Note that if 119899 = 119873then ignoring edge and vertex colouring the resulting graphis simply the 119873-cube Hence we term the correspondingAdinkra the119873-cube Adinkra (in Section 4 we show that thisis unique to119873) Where 119899 = 119873 we denote the correspondingAdinkra to be an (119873 119896) Adinkra where 119896 = 119873 minus 119899

When drawing Adinkra graphs we represent the bipar-tition by black and white vertices As in previous work byDoran et al [3] the height assignments will be representedby arranging the vertices in rows incrementally according toheight

Example 3 The Adinkras of Figure 1 satisfy all requirementslisted above Note that nodes in the same bipartition (bosons


(i) (ii)


















120601R120595)

where R120601and R




















⟨119906 V⟩ equiv119873

sum

119894=1119906119894V119894 (mod 2) (1)




1 1 1 1 0 0 0 00 0 1 1 1 1 0 00 0 0 0 1 1 1 1


1198623 rarr 1198622 rarr 1198628 to give ( 1 1 0 1 0 0 1 00 1 0 1 1 1 0 00 0 1 0 1 1 0 1

) We then perform







(000)

(100)

(110)

(111)

(011)

(010)(001)

(101)












120574119894 120574119895 = 120574

119894120574119895+ 120574119895120574119894= 21205751198941198951 (2)

where 120575119894119895




119894120574119895= minus120574119895120574119894for 119894 = 119895














119894









120594(119862)






1015840 hgt 120594 120587) with






119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896


(4)





(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(i) (ii)


















120601R120595)

where R120601and R




















⟨119906 V⟩ equiv119873

sum

119894=1119906119894V119894 (mod 2) (1)




1 1 1 1 0 0 0 00 0 1 1 1 1 0 00 0 0 0 1 1 1 1


1198623 rarr 1198622 rarr 1198628 to give ( 1 1 0 1 0 0 1 00 1 0 1 1 1 0 00 0 1 0 1 1 0 1

) We then perform







(000)

(100)

(110)

(111)

(011)

(010)(001)

(101)












120574119894 120574119895 = 120574

119894120574119895+ 120574119895120574119894= 21205751198941198951 (2)

where 120575119894119895




119894120574119895= minus120574119895120574119894for 119894 = 119895














119894









120594(119862)






1015840 hgt 120594 120587) with






119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896


(4)





(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























⟨119906 V⟩ equiv119873

sum

119894=1119906119894V119894 (mod 2) (1)




1 1 1 1 0 0 0 00 0 1 1 1 1 0 00 0 0 0 1 1 1 1


1198623 rarr 1198622 rarr 1198628 to give ( 1 1 0 1 0 0 1 00 1 0 1 1 1 0 00 0 1 0 1 1 0 1

) We then perform







(000)

(100)

(110)

(111)

(011)

(010)(001)

(101)












120574119894 120574119895 = 120574

119894120574119895+ 120574119895120574119894= 21205751198941198951 (2)

where 120575119894119895




119894120574119895= minus120574119895120574119894for 119894 = 119895














119894









120594(119862)






1015840 hgt 120594 120587) with






119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896


(4)





(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(000)

(100)

(110)

(111)

(011)

(010)(001)

(101)












120574119894 120574119895 = 120574

119894120574119895+ 120574119895120574119894= 21205751198941198951 (2)

where 120575119894119895




119894120574119895= minus120574119895120574119894for 119894 = 119895














119894









120594(119862)






1015840 hgt 120594 120587) with






119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896


(4)





(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119894









120594(119862)






1015840 hgt 120594 120587) with






119894=

(1198861 1198862 119886119873minus119896)

120587119894 (119909) equiv

119873minus119896

sum

119895=1119909119895+119909 sdot 119860

119894

equiv

119873minus119896

sum

119895=1119909119895+1199092 (1198861) + 1199093 (1198861 + 1198862) + sdot sdot sdot

+ 119909119873minus119896


(4)





(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(111) (111)

(110) (110)(101) (101)(011) (011)

(100) (100)(001) (001)

(000) (000)

(010) (010)








equiv 119909+119860119895+ 119894 (mod 2)

(6)



sum

119897=1119909119897+119909 sdot 119860

119895)


+(

119873minus119896

sum

119897=1119911119897+ 119911 sdot 119860

119895) (mod 2)

(7)






equiv10038161003816100381610038161003816119860119895minus 110038161003816100381610038161003816 +


12|119860 minus 1| |119860|

(mod 2)

(8)


equiv 1 (mod 2) (9)














2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2


2unique







(mod 2)

(10)














1015840 hgt 120594 120587) to be the





119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894(119909) denotes the




119894(119909) = 120587

1015840

119894(119910) and similarly 120587

119894(V sdot 119909) =

1205871015840






119894 where A

119894= (1198861 1198862 119886119873minus119896) If


119860119894=

119899

prod

119895=1120574119904119895 (11)


119894 we have

119875119894 (119909) = (119909 119909 sdot 1198861199041

(119909 sdot 1198861199041) sdot 1198861199042 119909 sdot 119860

119894) (12)



119894(119910) are equal


120587119894 (119909) = 120587

1015840

119894(119909) iff sum

119890isin119875119894(119909)

120587 (119890) equiv sum

119890isin119875119894(119910)

120587 (119890)

(mod 2) (13)


119904119899(119909 sdot 119860

119894) equiv


119895 1 le


sum

119890isin119875119894(119909)


+ 119886119873minus119896




(14)

Hence 120587119894(119909) equiv |119909| + 119909 sdot 119860

119894(mod 2) and 1205871015840

119894(119910) equiv 120587

119894(119909) +

(sum119890isin119875119894(119909)

120587(119890)minussum119890isin119875119894(119910)


120587(119890)+119909sdot119860119894

simplifies to

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| minus

119873minus119896

sum

119895=1(119909119895119886119895) (15)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv |119909| +

1003816100381610038161003816119860 1198941003816100381610038161003816 |119909| +

10038161003816100381610038161199101003816100381610038161003816 +1003816100381610038161003816119860 119894100381610038161003816100381610038161003816100381610038161199101003816100381610038161003816

+

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

(mod 2)

(16)


119894| + 1) equiv 0 (mod 2)


120587119894 (119909) minus 120587

1015840

119894(119910) equiv

119873minus119896

sum

119895=1(119909119895119886119895) +

119873minus119896

sum

119895=1(119910119895119886119895)

equiv ⟨119909 119860119894⟩ + ⟨119910 119860

119894⟩ (mod 2)

(17)














51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












51 Notation






119860 =

[[[[[[[[[[[[[[[[[

[

119894 0 0 0 1 2 3 0

0 119894 0 0 minus2 1 0 3

0 0 119894 0 minus3 0 1 minus2

0 0 0 119894 0 minus3 2 1


2 1 0 minus3 0 minus119894 0 0

3 0 1 2 0 0 minus119894 0

0 3 minus2 1 0 0 0 minus119894

]]]]]]]]]]]]]]]]]

]

(18)



1198711=

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

1198712=

[[[[[

[

0 1 0 0

minus1 0 0 0

0 0 0 minus1

0 0 1 0

]]]]]

]

1198713=

[[[[[

[

0 0 1 0

0 0 0 1

minus1 0 0 0

0 minus1 0 0

]]]]]

]

(19)


119894=

119871119879



120574119894= [

0 119871119894

1198771198940] (20)


Tr (119871119894(119871119895)119879

119871119896(119871119897)119879)

= 4 (120575119894119895120575119896119897minus 120575119894119896120575119895119897+ 120575119894119897120575119895119896+1205940120598119894119895119896119897)

(21)









times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












times





such that 119871119894= 119877119879




119872= (1198941 1198942 119894119905) represents







119909119909= 0 for some


proof

Corollary 29 119872119909119909



119909 isin 119881119872119909119909= 119872119910119910


119872











119905=


the sign of 119872119909119909


119894(119909) of Section 3)






119905minus1sum

119894=1(119901119894119909119894) + |119909| equiv |119909|

10038161003816100381610038161199011003816100381610038161003816 + ⟨119909 119901⟩

(mod 2)

(22)


119909119909= 119872119910119910

if andonly if ⟨119909 119901⟩ = ⟨119910 119901⟩







1198941119871119879

1198942sdot sdot sdot 119871119879

119894119905 Denoting path 119901 = (1198941 1198942 119894119905) we







119894119873 where 119901 =

(1198941 1198942 119894119873) equals

2119873minus119896minus1 (120590119901+1205940120598119901) (23)













120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120574119894= [

0 119871119894

1198771198940]

(minus1)119865= [

1 0

0 minus1]

(24)


119872 = [

1198711198941119871119879

1198942sdot sdot sdot 119871119879

1198941199050

0 119871119879

11989411198711198942sdot sdot sdot 119871119894119905

] (25)







Consider









119866



120583119866 (ℎ 119886) =

1003816100381610038161003816V isin119881 hgt (V) = ℎ 120591 (V) = 1198861003816100381610038161003816 (27)




119866(ℎ 119886) = 120583

119867(ℎ 119886) for all 1 le ℎ le 119905


cert119866 (V) = (120582V (119909) hgt (119909) 120591 (119909)) 119909 isin119881 (28)






119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119867(V) if and



119866(V) = cert

119867(V) Then since 119866 and119867


119867(V) there


119866(V) = cert

119867(V) hence 119866 and119867 are

isomorphic






120583119866 (3 1) = 2

120583119866 (3 2) = 0

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 2

120583119866 (1 2) = 4

120583119867 (3 1) = 0

120583119867 (3 2) = 2

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 4

120583119867 (1 2) = 2

(29)


119866= cert

119867 and the two







2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2119873

22119896+a|Aut (119862)| (30)





119862perp= V= (V1 V119899) isinZ

119899

2 ⟨V 119888⟩ = 0 forall119888 isin119862 (31)








120583119866 (3 1) = 1 (32)

120583119866 (3 2) = 1

120583119866 (2 1) = 120583119866 (2 2) = 4

120583119866 (1 1) = 3

120583119866 (1 2) = 3

120583119867 (3 1) = 2

120583119867 (3 2) = 0

120583119867 (2 1) = 120583119867 (2 2) = 4

120583119867 (1 1) = 2

120583119867 (1 2) = 4

(33)


Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Hence cert119866






120583119866 (3 1) = 120583119866 (3 2) = 1

120583119866 (2 119894) = 2 forall119894 isin [4]

120583119866 (1 1) = 120583119866 (1 2) = 1

120583119866 (1 3) = 120583119866 (1 4) = 2

120583119867 (3 2) = 2

120583119867 (2 119894) = 2 forall119894 isin [4]

120583119867 (1 2) = 0

120583119867 (1 1) = 120583119866 (1 3) = 120583119866 (1 4) = 2

(34)


= cert119867 and


7 Numerical Results









(V 1198941V 119894119873V 11989421198941V 1198941198731198941V 11989431198942V 119894119873119894119873minus1























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Acknowledgment



rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










rArr


rArr



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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