Algebra.pdf 2

23rd International Conference on

Formal Power Series and Algebraic Combinatorics

23e Conference Internationale sur les

Series Formelles et la Combinatoire Algebrique

Reykjavk, June 1317, 2011

13658050 c 2011 Discrete Mathematics and Theoretical Computer ScienceDMTCS, Nancy, France http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings

Program CommitteeMireille Bousquet-Melou (CNRS, Universite Bordeaux 1, France; chair)Michelle Wachs (University of Miami, USA; chair)Michael Albert (University of Otago, New Zealand)Matthias Beck (San Francisco State University, USA)Eva-Maria Feichtner (University of Bremen, Germany)Philippe Flajolet (INRIA Rocquencourt, France)Ira Gessel (Brandeis University, USA)Patricia Hersh (North Carolina State University, USA)Thomas Lam (University of Michigan, USA)Svante Linusson (KTH Royal Institute of Technology, Sweden)Ezra Miller (Duke University, USA)Isabella Novik (University of Washington, USA)Nathan Reading (North Carolina State University, USA)Andrew Rechnitzer (University of British Columbia, Canada)Astrid Reifegerste (University of Magdeburg, Germany)Yuval Roichman (Bar-Ilan University, Israel)Gilles Schaeffer (CNRS, Ecole Polytechnique, France)John Shareshian (Washington University in St. Louis, USA)Stephanie van Willigenburg (University of British Columbia, Canada)Vince Vatter (University of Florida, USA)Hiro-Fumi Yamada (Okayama University, Japan)Jiang Zeng (Universite Claude Bernard Lyon 1, France)

Organizing CommitteeEinar Steingrmsson (University of Strathclyde, UK; chair)Anders Claesson (University of Strathclyde, UK)Mark Dukes (University of Iceland, Iceland)Susanna Fishel (Arizona State University, USA)Axel Hultman (Linkoping University, Sweden)Sergey Kitaev (Reykjavk University, Iceland)Christopher Severs (Reykjavk University, Iceland)Henning Ulfarsson (Reykjavk University, Iceland)Ole Warnaar (University of Queensland, Australia)Mike Zabrocki (York University, Canada)

Proceedings compiled by Axel Hultman with translation assistance from Jean Betrema, MireilleBousquet-Melou and Lydie Mpinganzima.

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Table of Contents

Philippe Flajolet, the Father of Analytic Combinatorics vii

Invited talks ix

Contributed talks and posters

Marcelo Aguiar and Carlos Andre and Carolina Benedetti and Nantel Ber-geron and Zhi Chen and Persi Diaconis and Anders Hendrickson and SamuelHsiao and I. Martin Isaacs and Andrea Jedwab and Kenneth Johnson andGizem Karaali and Aaron Lauve and Tung Le and Stephen Lewis and Hu-ilan Li and Kay Magaard and Eric Marberg and Jean-Christophe Novelli andAmy Pang and Franco Saliola and Lenny Tevlin and Jean-Yves Thibon andNathaniel Thiem and Vidya Venkateswaran and C. Ryan Vinroot and NingYan and Mike ZabrockiSupercharacters, symmetric functions in noncommuting variables, extended abstract 3

Marcelo Aguiar and Aaron LauveLagranges Theorem for Hopf Monoids in Species 15

Federico Ardila and Thomas Bliem and Dido SalazarGelfandTsetlin Polytopes and FeiginFourierLittelmannVinberg Polytopes as MarkedPoset Polytopes 27

Drew ArmstrongHyperplane Arrangements and Diagonal Harmonics 39

Drew Armstrong and Brendon RhoadesThe Shi arrangement and the Ish arrangement 51

Jean-Christophe Aval and Adrien Boussicault and Philippe NadeauTree-like tableaux 63

J.-C. Aval and J.-C. Novelli and J.-Y. ThibonThe # product in combinatorial Hopf algebras 75

Cristina BallantinePowers of the Vandermonde determinant, Schur functions, and the dimension game 87

Jason Bandlow and Anne Schilling and Mike ZabrockiThe MurnaghanNakayama rule for k-Schur functions 99

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Nicholas R. Beaton and Filippo Disanto and Anthony J. Guttmann and Si-mone RinaldiOn the enumeration of column-convex permutominoes 111

Chris Berg and Nantel Bergeron and Sandeep Bhargava and Franco SaliolaPrimitive orthogonal idempotents for R-trivial monoids 123

Francois Bergeron and Nicolas Borie and Nicolas M. ThieryDeformed diagonal harmonic polynomials for complex reflection groups 135

Riccardo Biagioli and Fabrizio CaselliEnumerating projective reflection groups 147

Hoda BidkhoriFinite Eulerian posets which are binomial or Sheffer 159

Sara Billey and Andrew CritesRational smoothness and affine Schubert varieties of type A 171

Pavle V. M. Blagojevi and Benjamin Matschke and Gunter M. ZieglerA tight colored Tverberg theorem for maps to manifolds (extended abstract) 183

Saul A. BlancoShortest path poset of Bruhat intervals 191

Florian BlockRelative Node Polynomials for Plane Curves 199

Clemens Bruschek and Hussein Mourtada and Jan SchepersArc Spaces and Rogers-Ramanujan Identities 211

Anders Claesson and Mark Dukes and Martina KubitzkePartition and composition matrices: two matrix analogues of set partitions 221

Sam Clearman and Brittany Shelton and Mark SkanderaPath tableaux and combinatorial interpretations of immanants for class functions on Sn 233

Sylvie Corteel and Sandrine Dasse-HartautStatistics on staircase tableaux, eulerian and mahonian statistics 245

Maciej Doga and Piotr niadyPolynomial functions on Young diagrams arising from bipartite graphs 257

Art M. Duval and Caroline J. Klivans and Jeremy L. MartinCritical Groups of Simplicial Complexes 269

Richard Ehrenborg and JiYoon JungThe topology of restricted partition posets 281

Sergi ElizaldeAllowed patterns of -shifts 293

Alexander Engstrom and Patrik NorenPolytopes from Subgraph Statistics 305

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Valentin Feray and Piotr niadyDual combinatorics of zonal polynomials 317

Jeffrey FerreiraA Littlewood-Richardson type rule for row-strict quasisymmetric Schur functions 329

Alex Fink and David SpeyerK-classes for matroids and equivariant localization 339

Susanna Fishel and Eleni Tzanaki and Monica VaziraniCounting Shi regions with a fixed separating wall 351

Stefan Forcey and Aaron Lauve and Frank SottileCofree compositions of coalgebras 363

Benjamin IriarteDissimilarity Vectors of Trees and Their Tropical Linear Spaces (Extended Abstract) 375

Samuele GiraudoAlgebraic and combinatorial structures on Baxter permutations 387

Max GlickThe pentagram map and Y -patterns 399

Andrew Goodall and Criel Merino and Anna de Mier and Marc NoyOn the evaluation of the Tutte polynomial at the points (1,1) and (2,1) 411Alain Goupil and Hugo CloutierEnumeration of minimal 3D polyominoes inscribed in a rectangular prism 423

Alan GuoCyclic sieving phenomenon in non-crossing connected graphs 435

J. HaglundA polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants(condensed version) 445

Christopher R. H. Hanusa and Brant C. JonesThe enumeration of fully commutative affine permutations 457

Christine E. Heitsch and Prasad TetaliMeander Graphs 469

Gabor HetyeiThe short toric polynomial 481

Florent Hivert and Olivier MalletCombinatorics of k-shapes and Genocchi numbers 493

Jia Huang0-Hecke algebra actions on coinvariants and flags 505

Brandon Humpert and Jeremy L. MartinThe Incidence Hopf Algebra of Graphs 517

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IKEDA, Takeshi and NARUSE, Hiroshi and NUMATA, YasuhideBumping algorithm for set-valued shifted tableaux 527

Vt JelnekCounting self-dual interval orders 539

Miles Eli Jones and Jeffrey RemmelA reciprocity approach to computing generating functions for permutations with no pat-tern matches 551

Matthieu Josuat-Verge`s and Jang-Soo KimTouchard-Riordan formulas, T -fractions, and Jacobis triple product identity 563

Myrto Kallipoliti and Martina KubitzkeDouble homotopy Cohen-Macaulayness for the poset of injective words and the classicalNC-partition lattice 575

Matjaz KonvalinkaSkew quantum Murnaghan-Nakayama rule 587

Gilbert Labelle and Annie LacasseClosed paths whose steps are roots of unity 599

Carsten LangeMinkowski decompositions of associahedra 611

Matthias LenzHierarchical Zonotopal Power Ideals 623

Paul LevandeSpecial Cases of the Parking Functions Conjecture and Upper-Triangular Matrices 635

Joel Brewster Lewis and Ricky Ini Liu and Alejandro H. Morales and GretaPanova and Steven V Sam and Yan ZhangMatrices with restricted entries and q-analogues of permutations (extended abstract) 647

Sarah K Mason and Jeffrey RemmelRow-strict quasisymmetric Schur functions 657

Pierre-Loc MeliotKerovs central limit theorem for Schur-Weyl and Gelfand measures (extended abstract) 669

Alejandro H. Morales and Ekaterina A. VassilievaBijective evaluation of the connection coefficients of the double coset algebra 681

Gregg Musiker and Victor ReinerA topological interpretation of the cyclotomic polynomial 693

Suho OhGeneralized permutohedra, h-vectors of cotransversal matroids and pure O-sequences(extended abstract) 705

Suho Oh and Hwanchul YooTriangulations of n1 d1 and Tropical Oriented Matroids 717

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Masato Okado and Reiho SakamotoStable rigged configurations and LittlewoodRichardson tableaux 729

Kamilla Oliver and Helmut ProdingerHow often do we reject a superior value? Extended abstract 743

Greta PanovaTableaux and plane partitions of truncated shapes (extended abstract) 753

Adeline Pierrot and Dominique Rossin and Julian WestAdjacent transformations in permutations 765

Vincent Pilaud and Francisco SantosThe brick polytope of a sorting network 777

Svetlana PoznanoviCyclic sieving for two families of non-crossing graphs 789

Felipe RincnIsotropical Linear Spaces and Valuated Delta-Matroids 801

Vivien RipollSubmaximal factorizations of a Coxeter element in complex reflection groups 813

Dan RomikLocal extrema in random permutations and the structure of longest alternating subse-quences 825

Martin RubeyMaximal 0-1-fillings of moon polyominoes with restricted chain lengths and rc-graphs 835

Marc SageAsymptotics of several-partition Hurwitz numbers 849

Anne Schilling and Peter TingleyDemazure crystals and the energy function 861

Carsten SchultzThe equivariant topology of stable Kneser graphs 873

Luis Serrano and Christian StumpGeneralized triangulations, pipe dreams, and simplicial spheres 885

Armin StraubA q-analog of Ljunggrens binomial congruence 897

Nicholas TeffRepresentations on Hessenberg Varieties and Youngs Rule 903

Lenny TevlinNoncommutative Symmetric Hall-Littlewood Polynomials 915

Mirk VisontaiOn the monotone hook hafnian conjecture 927

vi

Philippe Flajolet, the Father of Analytic Combinatorics

Philippe Flajolet, mathematician and computer scientist extraordinaire, suddenly passed awayon March 22, 2011, at the prime of his career. He is celebrated for opening new lines of researchin analysis of algorithms, developing powerful new methods, and solving difficult open problems.His research contributions will have impact for generations, and his approach to research, basedon curiosity, a discriminating taste, broad knowledge and interest, intellectual integrity, and agenuine sense of camaraderie, will serve as an inspiration to those who knew him for years tocome.

The common theme of Flajolets extensive and far-reaching body of work is the scientificapproach to the study of algorithms, including the development of requisite mathematical andcomputational tools. During his forty years of research, he contributed nearly 200 publications,with an important proportion of fundamental contributions and representing uncommon breadthand depth. He is best known for fundamental advances in mathematical methods for the analysisof algorithms, and his research also opened new avenues in various domains of applied computerscience, including streaming algorithms, communication protocols, database access methods, datamining, symbolic manipulation, text-processing algorithms, and random generation. He exultedin sharing his passion: his papers had more than than a hundred different co-authors and he wasa regular presence at scientific meetings all over the world.

His research laid the foundation of a subfield of mathematics, now known as analytic combina-torics. His lifework Analytic Combinatorics (Cambridge University Press, 2009, co-authored withR. Sedgewick) is a prodigious achievement that now defines the field and is already recognizedas an authoritative reference.

Analytic combinatorics is a modern basis for the quantitative study of combinatorial struc-tures (such as words, trees, mappings, and graphs), with applications to probabilistic study ofalgorithms that are based on these structures. It also strongly influences other scientific domains,such as statistical physics, computational biology, and information theory. With deep historicroots in classical analysis, the basis of the field lies in the work of Knuth, who put the studyof algorithms on a firm scientific basis starting in the late 1960s with his classic series of books.Flajolets work takes the field forward by introducing original approaches in combinatorics basedon two types of methods: symbolic and analytic. The symbolic side is based on the automationof decision procedures in combinatorial enumeration to derive characterizations of generatingfunctions. The analytic side treats those functions as functions in the complex plane and leads toprecise characterization of limit distributions. In the last few years, Flajolet was further extendingand generalizing this theory into a meeting point between information theory, probability theoryand dynamical systems.

Philippe Flajolet was born in Lyon on December 1, 1948. He graduated from Ecole Poly-

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technique in Paris in 1970, and was immediately recruited as a junior researcher at the InstitutNational de Recherche en Informatique et Automatique (INRIA), where he spent his career. At-tracted by linguistics and logic, he worked on formal languages and computability with MauriceNivat, obtaining a PhD from the University of Paris 7 in 1973. Then, following Jean Vuillemin inthe footsteps of Don Knuth, he turned to the emerging field of analysis of algorithms and got aDoctorate in Sciences, both in mathematics and computer science, from the University of Paris atOrsay in 1979. At INRIA, he created and led the ALGO research group, which attracted visitingresearchers from all over the world.

He held numerous visiting positions, at Waterloo, Stanford, Princeton, Wien, Barcelona, IBMand the Bell Laboratories. He received several prizes, including the Grand Science Prize of UAP(1986), the Computer Science Prize of the French Academy of Sciences (1994), and the SilverMedal of CNRS (2004). He was elected a Corresponding Member (Junior Fellow) of the FrenchAcademy of Sciences in 1994, a Member of the Academia Europaea in 1995, and a Member(Fellow) of the French Academy of Sciences in 2003.

A brilliant, insightful honnete homme with broad scientific interests, Philippe pursued newdiscoveries in computer science and mathematics and shared them with students and colleaguesfor over 40 years with enthusiasm, joy, generosity, and warmth. In France, he was the majorreference at the interface between mathematics and computer science and founded the Aleameetings that bring together combinatorialists, probabilists and physicists to share problemsand methods involving discrete randomness. More broadly, he was the leading figure in thedevelopment of the international AofA (Analysis of Algorithms) community that is devoted toresearch on probabilistic, combinatorial, and asymptotic methods in the analysis of algorithms.The colleagues and students who are devoted to carrying on his work form the core of his primarylegacy.

May 2011Bruno Salvy, Bob Sedgewick, Miche`le Soria, Wojtek Szpankowski and Brigitte Vallee

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Invited talks

FPSAC: Flajolet, Power Series and AnalyticCombinatorics

Francois BergeronUniversite du Quebec a` Montreal, Canada

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The Law of AboavWeaire and its analogue inthree dimensions

Richard EhrenborgUniversity of Kentucky, USA

When investigating the structure of metals it is known that the atoms lie in a lattice structure.However, the lattice property only holds locally, that is, in a three dimensional cell called a grain.Bordering the grain is a boundary where the atoms lie chaotically, and beyond that is a newgrain where the lattice has a different orientation. The structure of these grains amounts to athree dimensional simple subdivision of space.

Looking at the two dimensional analogue, one observes that grains with a small number ofsides tend to be surrounded by grains with a large number of sides, and vice versa. The Lawof AboavWeaire states that the average number of sides of the neighbors of an n-sided grainshould be roughly 5 + 6/n. By introducing the correct error term we prove this law of MaterialScience and discuss its extension to three dimensions.

This is joint work with Menachem Lazar and Jeremy Mason. Moreover, selected work of vonNeumann, MacPherson and Srolovitz will be presented.

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Torus Squarings

Stefan FelsnerTechnical University of Berlin, Germany

A squaring is a tiling into squares of different sizes. In a seminal paper Brooks, Smith, Sto-ne and Tutte (1940) discussed squarings related to segment contact representations of planarquadrangulations. Regarding the squares of a squaring as vertices and edges as being definedby contacts we obtain the square dual graph. Schramm (1993) showed that 5-connected innertriangulations of a 4-gon can be represented as square duals. In this talk we review the planesituation and present some results concerning squarings of the torus and the graphs representedby them.

(joint work with E. Fusy)

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An alternative approach to alternating signmatrices

Ilse FischerUniversity of Vienna, Austria

Alternating sign matrices were first defined by Robbins and Rumsey in the early 1980s whenthey discovered that the -determinant, a natural generalization of the determinant, has anexpansion as a sum over all alternating sign matrices, just as the ordinary determinant has anexpansion as a sum over permutation matrices. Later it was observed that physicists had beenstudying a model for square ice that is equivalent to alternating sign matrices for a long time.Since then these square ice techniques are a standard tool to attack various enumeration problemsrelated to alternating sign matrices.

In my talk I shall present an alternative approach to alternating sign matrix enumerationwhich is more in the spirit of Zeilbergers original proof of the alternating sign matrix theorem.Starting point is an operator formula for the number of monotone triangles with prescribedbottom row. Refined enumerations of alternating sign matrices with respect to a fixed set ofboundary columns and rows can be expressed in terms of this operator formula. This enablesus to translate certain identities for the operator formula to identities for refined enumerationsof alternating sign matrices. This leads, on the one hand, to systems of linear equations thatdetermine the numbers uniquely, and, on the other hand, to surprisingly simple linear relationsbetween them. I will also report on recent attempts to translate these calculations into morecombinatorial reasonings.

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Generating functions for restricted Euleriannumbers

Ron GrahamUniversity of California, San Diego, USA

In this talk I will describe some recent work with Fan Chung on certain joint statistics forpermutations pi Sn. These involve the number of descents of pi, the maximum drop of pi andthe value of pi(n), and result in some new identities for restricted Eulerian numbers.

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The octahedral recurrence and generalizations

Richard KenyonBrown University, USA

This is joint work with A. Goncharov. The octahedral recurrence, or Hirota equation, is awell-known integrable discrete dynamical system, related to alternating sign matrices and dominotilings of Aztec diamonds.

We show that there is an underlying completely integrable Hamiltonian system commutingwith the octahedral recurrence. Formulas relating it to the octahedral recurrence can be writtenexplicitly in terms of dimers.

Similar systems exist for any periodic planar graph.

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Counting maps and graphs

Marc NoyPolytechnic University of Catalonia, Spain

The theory of map enumeration was started by Tutte in the 1960s, in an attempt to shed lighton the four colour problem, and since then the field has grown considerably. Many classes of mapshave been enumerated, including maps on surfaces, and connections have been found to otherfields, particularly to statistical physics. More recently, several classes of (unembedded) graphshave been analyzed, in particular planar graphs and graphs on surfaces, and precise asymptoticestimates have been obtained. In the talk we will review these results and the companion resultson the structure of random graphs from these families. The main tool in our work is analyticcombinatorics, as developed by Philippe Flajolet.

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Affine and projective tree metric theorems withapplications to phylogenetic reconstruction

Lior PachterUniversity of California, Berkeley, USA

The tree metric theorem provides a combinatorial four point condition that characterizesdissimilarity maps derived from pairwise compatible split systems. A similar (but weaker) fourpoint condition characterizes dissimilarity maps derived from circular split systems (Kalmansonmetrics). The tree metric theorem was first discovered in the context of phylogenetics and formsthe basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first consideredby computer scientists, and are notable in that they are a non-trivial class of metrics for which thetraveling salesman problem is tractable. We will review these theorems via the unifying frameworkof the (tropical) space of trees and extensions to PQ- and PC-trees, and will discuss applicationsto phylogenetic reconstruction. In particular, we will provide an explanation for the peculiar formof the balanced minimum evolution criterion popular in phylogenetics. The work to be presentedis joint with Aaron Kleinman.

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Simple KLR modules

Monica VaziraniUniversity of California at Davis, USA

Khovanov-Lauda-Rouquier (KLR) algebras have played a fundamental role in categorifyingquantum groups. I will discuss the structure of their simple modules, in particular that they carrythe structure of a crystal graph. This is joint work with Aaron Lauda.

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KP solitons, total positivity, and cluster algebras

Lauren WilliamsUniversity of California, Berkeley, USA

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev andPetviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as theKP equation. It is well-known that the Wronskian approach to the KP equation provides a methodto construct soliton solutions. More recently, several authors have focused on understanding theregular soliton solutions that one obtains in this way: these come from points of the totallynon-negative Grassmannian.

In joint work with Yuji Kodama, we establish a tight connection between Postnikovs theoryof total positivity for the Grassmannian, and the structure of regular soliton solutions to the KPequation. This connection allows us to apply machinery from total positivity to KP solitons. Inparticular, we classify the soliton graphs coming from the totally non-negative Grassmannian,when the absolute value of the time parameter is sufficiently large. We demonstrate an intriguingconnection between soliton graphs and the cluster algebras of Fomin and Zelevinsky. Finally, weapply this connection towards the inverse problem for KP solitons.

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Contributed talks and posters

FPSAC 2011, Reykjavk, Iceland DMTCS proc. AO, 2011, 314

Supercharacters, symmetric functions innoncommuting variables, extended abstract

Marcelo Aguiar and Carlos Andre and Carolina Benedetti and NantelBergeron and Zhi Chen and Persi Diaconis and Anders Hendrickson andSamuel Hsiao and I. Martin Isaacs and Andrea Jedwab and Kenneth John-son and Gizem Karaali and Aaron Lauve and Tung Le and Stephen Lewisand Huilan Li and Kay Magaard and Eric Marberg and Jean-ChristopheNovelli and Amy Pang and Franco Saliola and Lenny Tevlin and Jean-Yves Thibon and Nathaniel Thiem and Vidya Venkateswaran and C. RyanVinroot and Ning Yan and Mike Zabrocki

Abstract. We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis onthe group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functionsin noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developmentsin each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorialHopf algebras.

Resume. Nous montrons que deux structures en apparence bien differentes peuvent etre identifiees: les super-caracte`res, qui sont un outil commode pour faire de lanalyse de Fourier sur le groupe des matrices unipotentestriangulaires superieures a` coefficients dans un corps fini, et lanneau des fonctions symetriques en variables non-commutatives. Ces deux structures sont des alge`bres de Hopf isomorphes. Cette identification permet de traduiredans une structure les developements concus pour lautre, et sugge`re de nombreux exemples dans le domaine nou-veau des alge`bres de Hopf combinatoires.

Keywords: supercharacters, set partitions, symmetric functions in non-commuting variables, Hopf algebras

1 IntroductionIdentifying structures in seemingly disparate fields is a basic task of mathematics. An example, withparallels to the present work, is the identification of the character theory of the symmetric group withsymmetric function theory. This connection is wonderfully exposited in Macdonalds book [20]. Later,Geissinger and Zelevinsky independently realized that there was an underlying structure of Hopf algebrasthat forced and illuminated the identification [14, 27]. We present a similar program for a supercharactertheory associated to the uppertriangular group and the symmetric functions in noncommuting variables.

13658050 c 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

4 AABBCDHHIJJKLLLLMMNPSTTTVVYZ

Let UTn(q) be the group of uppertriangular matrices with entries in the finite field Fq and ones on thediagonal. This group is a Sylow p-subgroup of GLn(q). Describing the conjugacy classes or charactersof UTn(q) is a provably wild problem. In a series of papers, Andre developed a cruder theory thatlumps together various conjugacy classes into superclasses and considers certain sums of irreduciblecharacters as supercharacters. The two structures are compatible (so supercharacters are constant onsuperclasses). The resulting theory is very nicely behaved there is a rich combinatorics describinginduction and restriction along with an elegant formula for the values of supercharacters on superclasses.The combinatorics is described in terms of set partitions (the symmetric group theory involves integerpartitions) and the combinatorics seems akin to tableau combinatorics. At the same time, supercharactertheory is rich enough to serve as a substitute for ordinary character theory in some problems [6] .

In more detail, the group UTn(q) acts on both sides of the algebra of strictly upper-triangular matricesnn (which can be thought of as nn = UTn(q) 1). The two sided orbits on nn can be mapped backto UTn(q) by adding the identity matrix. These orbits form the superclasses in UTn(q). A similarconstruction on the dual space nn gives a collection of class functions on UTn(q) that turn out to beconstant on superclasses. These orbit sums (suitably normalized) are the supercharacters. Let

SC =n0

SCn,

where SCn is the set of functions from UTn(q) to C that are constant on superclasses, and SC0 =C-span{1} is by convention the set of class functions of UT0(q) = {}.

Let =

n0

n

be the ring of symmetric functions in non-commuting variables. Such functions were considered by Wolf[25] and Doubilet [12]. More recent work of Sagan brought them to the forefront. A lucid introduction isgiven by Rosas and Sagan [22] and combinatorial applications by Gebhard and Sagan [13]. The algebra is actively studied as part of the theory of combinatorial Hopf algebras [3, 7, 9, 10, 17, 21]. The mand thus are invariant under permutations of variables.

Our main result is to show that when q = 2, SC has a Hopf structure isomorphic to that of . Thisconstruction of a Hopf algebra from the representation theory of a sequence of groups is the main contri-bution of this paper. It differs from previous work in that supercharacters are used. Previous work wasconfined to ordinary characters (e.g. [19]) and the results of [8] indicate that this is a restrictive setting.This work opens the possibility for a vast new source of Hopf algebras.

AcknowledgementsThis paper developed during a focused research week at the American Institute of Mathematics in May2010. The main results presented here were proved as a group during that meeting.

2 Background2.1 Supercharacter theorySupercharacters were first studied by Andre (e.g. [4]) and Yan [26] in relation to UTn(q) in order tofind a more tractable way to understand the representation theory of UTn(q). Diaconis and Isaacs [11]

Supercharacters and Hopf algebras 5

then generalized the concept to arbitrary finite groups, and we reproduce a version of this more generaldefinition below.

A supercharacter theory of a finite group G is a pair (K,X ) where K is a partition of G and X is apartition of the irreducible characters of G such that

(a) Each K K is a union of conjugacy classes,

(b) {1} K, where 1 is the identity element of G, and {11} X , where 11 is the trivial character of G.

(c) For X X , the character X

(1)

is constant on the parts of K,

(d) |K| = |X |.

We will refer to the parts of K as superclasses, and for some fixed choice of scalars cX Q (which arenot uniquely determined), we will refer to the characters

X = cXX

(1), for X X

as supercharacters (the scalars cX should be picked such that the supercharacters are indeed characters).For more information on the implications of these axioms, including some redundancies in the definition,see [11].

There are a number of different known ways to construct supercharacter theories for groups, including

Gluing together group elements and irreducible characters using outer automorphisms [11],

Finding normal subgroups N / G and grafting together superchararacter theories for the normalsubgroup N and for the factor group G/N to get a supercharacter theory for the whole group [16].

This paper will however focus on a technique first introduced for algebra groups [11], and then generalizedto some other types of groups by Andre and Neto (e.g. [5]).

The group UTn(q) has a natural two-sided action on the Fq-spaces

n = UTn(q) 1 and n = Hom(n,Fq)

given by left and right multiplication on n and for n,

(uv)(x 1) = (u1(x 1)v1), for u, v, x UTn(q).

It can be shown that the orbits of these actions parametrize the superclasses and supercharacters, respec-tively, for a supercharacter theory. In particular, two elements u, v UTn(q) are in the same superclassif and only if u 1 and v 1 are in the same two-sided orbit in UTn(q)\n/UTn(q). Since the action


of UTn(q) on n can be viewed as applying row and column operations, we obtain a parameterization ofsuperclasses given by

{Superclassesof UTn(q)

}

u 1 n with at mostone nonzero entry ineach row and column .

This indexing set is central to the combinatorics of this paper, so we give several interpretations for it. Let

Sn(q) ={

Sets of triples i a_j = (i, j, a) [n] [n] Fq ,with i < j, and i a_j, k b_l implies i 6= k, j 6= l

},

where we will refer to the elements of Sn(q) as Fq -set partitions. In particular,

Sn(q) u 1 n with at mostone nonzero entry ineach row and column

= {i(i,j)_ j | (i, j) D} 7

ia_j

aeij ,

(2.1)

where eij is the matrix with 1 in the (i, j) position and zeroes elsewhere.

Remark 1 Consider the map

pi : Sn(q) Sn(2) 7 {i 1_j | i a_j }, (2.2)

which ignores the part of the data that involves field scalars. Note that Sn(2) is in bijection with the setpartitions of the set {1, 2, . . . , n}. Indeed, the connected components of an element Sn(2) may beviewed as the blocks of a partition of {1, 2, . . . , n}. Composing the map pi with this bijection associates aset partition to an element ofMn(q) or Sn(q), which we call the underlying set partition.

Fix a nontrivial homomorphism : F+q C. For each n, construct a UTn(q)-module

V = C-span{v | UTn(q) }

with left action given by

uv = ((u1 1))vu, for u UTn(q), UTn(q).

Up to isomorphism, these modules depend only on the two-sided orbit in UTn(q)\n/UTn(q) of .Furthermore, there is an injective function : Sn(q) n given by

() : n FqX 7

ia_j

aXij


that maps Sn(q) onto a natural set of orbit representatives in n. We will identify Sn(q) with() n.

The traces of the modules V for Sn(q) are the supercharacters of UTn(q), and they have a nicesupercharacter formula given by

(u) =

q#{(i,j,k)|i


by reordering the rows and columns according to (2.5). For example, if J = (14|3|256), then

UTJ(q) 3

1 0 0 a 0 00 1 0 0 b c0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 d0 0 0 0 0 1

stJ7((

1 a0 1

), (1),

(1 b c0 1 d0 0 1

)) UT2(q)UT1(q)UT3(q).

Combinatorially, if J = (J1|J2| |J`) we letSJ(q) = { Sn(q) | i a_j implies i, j are in the same part in J}.

Then we obtain the bijection

stJ : SJ(q) S|J1|(q) S|J2|(q) S|J`|(q) (2.7)that relabels the indices using the straightening map (2.5). For example, if J = 14|3|256, then

stJ

( 1 2 3 4 5 6

ab )

= 1 2

a 1

1 2 3

b

Note that UTm(q) UTn(q) is an algebra group, so it has a supercharacter theory with the standardconstruction [11] such that

SC(UTm(q)UTn(q)) = SCm SCn.The combinatorial map (2.7) preserves supercharacters across this isomorphism.

The first operation of interest is restriction

JResUTn(q)stJ (UTJ (q)) : SCn SC|J1| SC|J2| SC|J`| 7 ResUTn(q)UTJ (q)() st

1J ,

orJResUTn(q)stJ (UTJ (q))()(u) = (st

1J (u)), for u UT|J1|(q) UT|J`|(q).

Remark 2 There is an algorithmic method for computing restrictions of supercharacters (and also tensorproducts of characters) [23, 24]. This has been implemented in Sage.

For an integer composition (m1,m2, . . . ,m`) of n, let

UT(m1,m2,...,m`)(q) = UT(1,...,m1|m1+1,...,m1+m2||nm`+1,...,n)(q) UTm1++m`(q).There is a surjective homomorphism : UTn(q) UT(m1,m2,...,m`)(q) such that 2 = ( fixes thesubgroup UT(m1,m2,...,m`)(q) and sends the normal complement to 1). We now obtain the inflation map

InfUTn(q)UT(m1,m2,...,m`)(q): SCm1 SCm2 SCm` SCn,

whereInfUTn(q)UT(m1,m2,...,m`)(q)

()(u) = ((u)), for u UTn(q).


2.3 The Hopf algebra Symmetric polynomials in a set of commuting variablesX are the invariants of the action of the symmetricgroup SX of X by automorphisms of the polynomial algebra K[X] over a field K.

WhenX = {x1, x2, . . .} is infinite, we let SX be the set of bijections onX with finitely many nonfixedpoints. Then the subspace of K[[X]]SX of formal power series with bounded degree is the algebra ofsymmetric functions Sym(X) over K. It has a natural bialgebra structure defined by

(f) =k

f k f k , (2.8)

where the f k, fk are defined by the identity

f(X +X ) =k

f k(X)f k (X

), (2.9)

and X + X denotes the disjoint union of two copies of X . The advantage of defining the coproduct inthis way is that is clearly coassociative and that it is obviously a morphism for the product. For eachinteger partition = (1, 2, . . . , `), the monomial symmetric function corresponding to is the sum

m(X) =

xO(x)x (2.10)

over elements of the orbit O(x) of x = x11 x22 x`` under SX , and the monomial symmetric func-

tions form a basis of Sym(X). The coproduct of a monomial function is

(m) =

=m m . (2.11)

The dual basis m of m is a multiplicative basis of the graded dual Sym, which turns out to be iso-

morphic to Sym via the identification mn = hn (the complete homogeneous function, the sum of allmonomials of degree n).

The case of noncommuting variables is very similar. LetA be an alphabet, and consider the invariants ofSA acting by automorphisms on the free algebra KA. Two words a = a1a2 an and b = b1b2 bnare in the same orbit whenever ai = aj if and only if bi = bj . Thus, orbits are parametrized by setpartitions in at most |A| blocks. Assuming as above that A is infinite, we obtain an algebra based on allset partitions, defining the monomial basis by

m(A) =wO

w, (2.12)

where O is the set of words such that wi = wj if and only if i and j are in the same block of .One can introduce a bialgebra structure by means of the coproduct

(f) =k

f k f k where f(A +A) =k

f k(A)f k (A

), (2.13)


and A + A denotes the disjoint union of two mutually commuting copies of A. The coproduct of amonomial function is

(m) =

J[`()]mst(J ) mst(Jc ). (2.14)

This coproduct is cocommutative. With the unit that sends 1 to m and the counit (f(A)) = f(0, 0, . . .),we have that is a connected graded bialgebra and therefore a graded Hopf algebra.

Remark 3 We note that is often denoted in the literature as NCSym or WSym.

3 A Hopf algebra realization of SCIn this section we describe a Hopf structure for the space

SC =n0

SCn

= C-span{ | Sn(q), n Z0}= C-span{ | Sn(q), n Z0}.

The product and coproduct are defined representation theoretically by the inflation and restriction opera-tions of Section 2.2,

= InfUTa+b(q)UT(a,b)(q)( ), where SCa, SCb, (3.1)

and() =

J=(A|Ac)A[n]

JResUTn(q)UT|A|(q)UT|Ac|(q)(), for SCn. (3.2)

For a combinatorial description of the Hopf structure of SC it is most convenient to work with the super-class characteristic functions.

Proposition 3.1

(a) For Sk(q), Snk(q),

=

=unionsqunionsq(k+)Sn(q)ia_l implies ik


Example 1 We have

1 2 3

a 1 2 3 4

b c = 1 2 3 4 5 6 7

a b c +dFq

( 1 2 3 4 5 6 7

a d b c + 1 2 3 4 5 6 7

a d b c + 1 2 3 4 5 6 7

adb c +

1 2 3 4 5 6 7

ad

b c

)

+

d,eFq

( 1 2 3 4 5 6 7

ad

e b c + 1 2 3 4 5 6 7

a deb c

).

and

( 1 2 3 4

a

)= 1 2 3 4

a + 2 1 2 3

a 1

+ 1 2

a 1 2

+ 1 2

1 2

a + 21

1 2 3

a + 1 2 3 4

a .

By comparing Proposition 3.1 to (2.14) and the product on monomials, we obtain the following theo-rem.

Theorem 3.2 For q = 2, the mapch : SC

7 mis a Hopf algebra isomorphism.

Note that although we did not assume for the theorem that SC is a Hopf algebra, the fact that chpreserves the Hopf operations implies that SC for q = 2 is indeed a Hopf algebra.

Corollary 3.3 The algebra SC with product given by (3.1) and coproduct given by (3.2) is a Hopf alge-bra.

Remark 4

(a) Note that the isomorphism of Theorem 3.2 is not in any way canonical. In fact, the automorphismgroup of is rather large, so there are many possible isomorphisms. For our chosen isomor-phism, there is no nice interpretation for the image of the supercharacters under the isomorphismof Theorem 3.2. Even less pleasant, when one composes it with the map

Sym

that allows variables to commute (see [12, 22]), one in fact obtains that the supercharacters arenot Schur positive. But, exploration with Sage suggests that it may be possible to choose an isomor-phism such that the image of the supercharacters are Schur positive.

(b) Although the antipode is determined by the bialgebra structure of , explicit expressions are notwell understood. However, there are a number of forthcoming papers (e.g. [2, 18]) addressing thissituation.


The Hopf algebra SC has a number of natural Hopf subalgebras. One of particular interest is thesubspace spanned by linear characters (characters with degree 1). In fact, for this supercharacter theoryevery linear character of Un is a supercharacter and by (2.4) these are exactly indexed by the set

Ln = { Sn(q) | i a_j implies j = i+ 1}.

Corollary 3.4 For q = 2, the Hopf subalgebra

LSC = C-span{ | i 1_j implies j = i+ 1},

is isomorphic to the Hopf algebra of noncommutative symmetric functions Sym studied in [15].

Remark 5 In fact, for each k Z0 the space

SC(k) = C-span{ | i_j implies j i k}

is a Hopf subalgebra. This gives an unexplored filtration of Hopf algebras which interpolate betweenLSC and SC.

References[1] M. Aguiar, N. Bergeron and F. Sottile, Combinatorial Hopf Algebras and generalized Dehn

Sommerville relations, Compositio Math. 142 (2006), 130.

[2] M. Aguiar, N. Bergeron and N. Thiem, A Hopf monoid from the representation theory of the finitegroup of unitriangular matrices, in preparation.

[3] M. Aguiar and S. Mahajan, Coxeter groups and Hopf algebras, Amer. Math. Soc. Fields Inst.Monogr. 23 (2006).

[4] C. Andre, Basic characters of the unitriangular group, J. Algebra 175 (1995), 287319.

[5] C. Andre and A Neto, Super-characters of finite unipotent groups of types Bn, Cn and Dn, J.Algebra 305 (2006), 394429.

[6] E. Arias-Castro, P. Diaconis and R. Stanley, A super-class walk on upper-triangular matrices, J.Algebra 278 (2004), 739765.

[7] N. Bergeron, C. Hohlweg, M. Rosas and M. Zabrocki, Grothendieck bialgebras, partition lattices,and symmetric functions in noncommutative variables, Electron. J. Combin. 13 (2006), no. 1, Re-search Paper 75, 19 pp.

[8] N. Bergeron, T. Lam and H. Li, Combinatorial Hopf algebras and Towers of Algebras Dimension,Quantization and Functorality, Preprint arXiv:0710.3744v1.

[9] N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and coinvariants of the sym-metric groups in noncommuting variables, Canad. J. Math. 60 (2008), no. 2, 266296.


[10] N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasi-symmetricfunctions in non-commutative variables are free and co-free, J. Algebra Appl. 8 (2009), no. 4,581600. Commun. Math. Phys. 199 (1998) 203242.

[11] P. Diaconis and M Isaacs, Supercharacters and superclasses for algebra groups, Trans. Amer. Math.Soc. 360 (2008), 23592392.

[12] P. Doubilet, On the foundations of combinatorial theory. VII: Symmetric functions through thetheory of distribution and occupancy, Studies in Applied Math. 51 (1972), 377396.

[13] D.D. Gebhard and B.E. Sagan, A chromatic symmetric function in noncommuting variables, J.Algebraic Combin. 13 (2001), no. 3, 227255.

[14] L. Geissinger, Hopf algebras of symmetric functions and class functions, Lecture Notes in Math.579 (1977) 168181.

[15] I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon, Noncommutativesymmetric functions, Adv. Math. 112 (1995), 218348.

[16] A. Hendrickson, Supercharacter theories of finite cyclic groups. Unpublished Ph.D. Thesis, Depart-ment of Mathematics, University of Wisconsin, 2008.

[17] F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, J. AlgebraicCombin. 28 (2008), no. 1, 6595.

[18] A. Lauve and M. Mastnak, The primitives and antipode in the Hopf algebra of symmetric functionsin noncommuting variables, Preprint arXiv:1006.0367.

[19] H. Li, Algebraic Structures of Grothendieck Groups of a Tower of Algebras, Ph.D. Thesis, YorkUniversity, 2007.

[20] I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press,1995.

[21] J.-C. Novelli and J.-Y. Thibon, Polynomial realizations of some trialgebras, FPSAC06. Alsopreprint ArXiv:math.CO/0605061.

[22] M.H. Rosas and B.E. Sagan, Symmetric functions in noncommuting variables, Trans. Amer. Math.Soc. 358 (2006), no. 1, 215232

[23] N. Thiem, Branching rules in the ring of superclass functions of unipotent upper-triangular matri-ces, J. Algebraic Combin. 31 (2010), no. 2, 267298.

[24] N. Thiem and V. Venkateswaran, Restricting supercharacters of the finite group of unipotent upper-triangular matrices, Electron. J. Combin. 16(1) Research Paper 23 (2009), 32 pages.

[25] M.C. Wolf, Symmetric functions of non-commuting elements, Duke Math. J. 2 (1936) 626637.

[26] N. Yan, Representation theory of the finite unipotent linear groups, Unpublished Ph.D. Thesis,Department of Mathematics, University of Pennsylvania, 2001.


[27] A. Zelevinsky. Representations of Finite Classical Groups, Springer Verlag, 1981.

M. Aguiar, University of Texas A&M,C. Andre, University of Lisbon,C. Benedetti, York University,N. Bergeron, York University, supported by CRC and NSERC,Z. Chen, York University,P. Diaconis, Stanford University, supported by NSF DMS-0804324,A. Hendrickson, Concordia College,S. Hsiao, Bard College,I.M. Isaacs, University of Wisconsin-Madison,A. Jedwab, University of Southern California, supported by NSF DMS 07-01291,K. Johnson, Penn State Abington,G. Karaali, Pomona College,A. Lauve, Loyola University,T. Le, University of Aberdeen,S. Lewis, University of Washington, supported by NSF DMS-0854893,H. Li, Drexel University, supported by NSF DMS-0652641,K. Magaard, University of Birmingham,E. Marberg, MIT, supported by NDSEG Fellowship,J.-C. Novelli, Universite Paris-Est Marne-la-Vallee,A. Pang, Stanford University,F. Saliola, Universite du Quebec a Montreal, supported by CRC,L. Tevlin, New York University,J.-Y. Thibon, Universite Paris-Est Marne-la-Vallee,N. Thiem, University of Colorado at Boulder, supported by NSF DMS-0854893,V. Venkateswaran, Caltech University,C.R. Vinroot, College of William and Mary, supported by NSF DMS-0854849,N. Yan,M. Zabrocki, York University.


Lagranges Theorem for Hopf Monoids inSpecies

Marcelo Aguiar1and Aaron Lauve2

1Department of Mathematics, Texas A&M University, College Station, TX, USA2Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA

Abstract. We prove Lagranges theorem for Hopf monoids in the category of connected species. We deduce necessaryconditions for a given subspecies k of a Hopf monoid h to be a Hopf submonoid: each of the generating seriesof k must divide the corresponding generating series of k in N[[x]]. Among other corollaries we obtain necessaryinequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In theset-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence.

Resume. Nous prouvons le theore`me de Lagrange pour les monodes de Hopf dans la categorie des espe`ces connexes.Nous deduisons des conditions necessaires pour quune sous-espe`ce k dun monode de Hopf h soit un sous-monodede Hopf: chacune des series generatrices de k doit diviser la serie generatrice correspondante de h dans N[[x]]. Parmidautres corollaires nous trouvons des inegalites necessaires pour quune suite dentiers soit la suite des dimensionsdun monode de Hopf. Dans le cas ensembliste les inegalites sont lineaires et exigent que la transformee binomialede la suite soit non negative.

Keywords: Hopf monoids, species, graded Hopf algebras, Lagranges theorem, generating series

IntroductionLagranges theorem states that for any subgroup K of a group H , H = K Q as (left) K-sets, whereQ = H/K. In particular, if H is finite, |K| divides |H|. Passing to group algebras over a field k, we havethat kH = kKkQ as (left) kK-modules, or that kH is free as a kK-module. Kaplansky [6] conjecturedthat the same statement holds for Hopf algebras (group algebras being principal examples). It turns outthat the result does not not in general, as shown by Oberst and Schneider [13, Proposition 10] and [11,Example 3.5.2]. On the other hand, the result does hold for large classes of Hopf algebras, includingthe finite dimensional ones by a theorem Nichols and Zoeller [12], and the pointed ones by a theorem ofRadford [16]. More information can be found in Sommerhausers survey [15].

The main result of this paper (Theorem 7) is a version of Lagranges theorem for Hopf monoids in thecategory of connected species. (Hopf algebras are Hopf monoids in the category of vector spaces.) Animmediate application is a test for Hopf submonoids (Corollary 12): if any one of the generating series for

Aguiar supported in part by NSF grant DMS-1001935. [email protected] supported in part by NSA grant H98230-10-1-0362. [email protected]


16 Marcelo Aguiar and Aaron Lauve

a subspecies k does not divide the corresponding generating series for the Hopf monoid h (in the sensethat the quotient has negative coefficients), then k is not a Hopf submonoid of h. A similar test also holdsfor connected graded Hopf algebras (Corollary 4). The proof of Theorem 7 (for Hopf monoids in species)parallels Radfords proof (for Hopf algebras).

This abstract is organized as follows. In Section 1, we recall Lagranges theorem for several classesof Hopf algebras. In Section 2, we recall the basics of Hopf monoids in species and prove Lagrangestheorem in this setting. We conclude in Section 3 with some examples and applications involving theassociated generating series. Among these, we derive necessary conditions for a sequence of nonnegativeintegers to be the sequence of dimensions of a connected Hopf monoid in species.

1 Lagranges theorem for Hopf algebrasWe begin by recalling a couple of versions of this theorem. (All vector spaces are over a fixed field k.)

Theorem 1 LetH be a finite dimensional Hopf algebra over a field k. IfK H is any Hopf subalgebra,then H is a free left (and right) K-module.

This is the Nichols-Zoeller theorem [12]; see also [11, Theorem 3.1.5]. We will not make direct use ofthis result, but rather the related results discussed below.

A Hopf algebra H is pointed if all its simple subcoalgebras are one dimensional. Equivalently, thegroup-like elements of H linearly span the coradical of H . Given a subspace K of H , let

K+ := K ker()

where : H k is the counit of H . Also, K+H denotes the right H-ideal generated by K+.

Theorem 2 Let H be a pointed Hopf algebra. If K H is any Hopf subalgebra, then H is a free left(and right) K-module. Moreover, H = K (H/K+H) as left K-modules.The first statement is due to Radford [16, Section 4] and the second (stronger) statement is due to Schnei-der [14, Remark 4.14]. See Sommerhausers survey [15] for further generalizations.

A Hopf algebra H is graded if there is given a decomposition

H =n0

Hn

into linear subspaces that is preserved by all operations. It is connected if in addition H0 is linearlyspanned by the unit element.

Theorem 3 Let H be a graded connected Hopf algebra. If K H is a graded Hopf subalgebra, then His a free left (and right) K-module. Moreover,

H = K (H/K+H)

as left K-modules and as graded vector spaces.

Lagranges Theorem 17

Proof: Since H is connected graded, its coradical is H0 = k, so H is pointed and Theorem 2 applies.Radfords proof shows that there exists a graded vector space Q such that

H = K Qas leftK-modules and as graded vector spaces. (The argument we give in the parallel setting of Theorem 7makes this clear.) Note that K+ =

n1Kn, and K+H and H/K+H inherit the grading of H . To

complete the proof, it suffices to show that Q = H/K+H as graded vector spaces. 2Given a graded Hopf algebraH , let PH(x) N[[x]] denote its Poincare seriesthe formal power series

enumerating the dimensions of its graded components,

PH(x) :=n0

dimHn xn.

Suppose H is graded connected and K is a graded Hopf subalgebra. In this case, their Poincare series areof the form 1 + a1x + a2x2 + with ai N and the quotient PH(x)/PK(x) is a well-defined powerseries in Z[[x]].

Corollary 4 Let H be a connected graded Hopf algebra. If K H is any graded Hopf subalgebra, thenthe quotient PH(x)/PK(x) of Poincare series is nonnegative, i.e., belongs to N[[x]].

Proof: By Theorem 3, H = K Q as graded vector spaces, where Q = H/K+H . Hence PH(x) =PK(x)PQ(x) and the result follows. 2

Example 5 Consider the Hopf algebra QSym of quasisymmetric functions in countably many variables,and the Hopf subalgebra Sym of symmetric functions. They are graded connected, so by Theorem 3,QSym is a free module over Sym. Garsia and Wallach prove this same fact for the algebras QSymn andSymn of (quasi) symmetric functions in n variables [4]. These are not Hopf algebras when n is finite, soTheorem 3 does not yield the result of Garsia and Wallach. The papers [4] and [8] provide information onthe space Qn entering in the decomposition QSymn = Symn Qn.

2 Lagranges theorem for Hopf monoids in speciesWe first review the notion of Hopf monoid in the category of species, following [2], and then proveLagranges theorem in this setting. We restrict attention to the case of connected Hopf monoids.

2.1 Hopf monoids in speciesThe notion of species was introduced by Joyal [5]. It formalizes the notion of combinatorial structure andprovides a framework for studying the generating functions which enumerate these structures. The book[3] by Bergeron, Labelle and Leroux expounds the theory of (set) species.

Joyals work indicates that species may also be regarded as algebraic objects; this is the point of viewadopted in [2] and in this work. To this end, it is convenient to work with vector species.

A (vector) species is a functor q from finite sets and bijections to vector spaces and linear maps.Specifically, it is a family of vector spaces q[I], one for each finite set I , together with linear mapsq[] : q[I] q[J ], one for each bijection : I J , satisfying

q[idI ] = idq[I] and q[ ] = q[] q[ ]


whenever and are composable bijections. A species q is finite dimensional if each vector space q[I]is finite dimensional. In this paper, all species are finite dimensional. A morphism of species is a naturaltransformation of functors. Let Sp denote the category of (finite dimensional) species.

We give two elementary examples that will be useful later.

Example 6 Let E be the exponential species, defined by E[I] = k{I} for all I . The symbol I denotesan element canonically associated to the set I (for definiteness, we may take I = I). Thus, E[I] is a onedimensional space with a distinguished basis element. A richer example is provided by the species L oflinear orders, defined by L[I] = k{linear orders on I} for all I (a space of dimension n! when |I| = n).

We use to denote the Cauchy product of two species. Specifically,(p q)[I] :=

SunionsqT=Ip[S] q[T ] for all finite sets I.

The notation S unionsq T = I indicates that S T = I and S T = . The sum runs over all such ordereddecompositions of I , or equivalently over all subsets S of I: there is one term for S unionsq T and another forT unionsq S. The Cauchy product turns Sp into a symmetric monoidal category. The braiding simply switchesthe tensor factors. The unit object is the species 1 defined by

1[I] :=

{k if I is empty,0 otherwise.

A monoid in the category (Sp, ) is a species m together with a morphism of species : m mm,i.e., a family of maps

S,T : m[S]m[T ]m[I],one for each ordered decomposition I = S unionsqT , satisfying appropriate associativity and unital conditions,and naturality with respect to bijections. Briefly, to each m-structure on S and m-structure on T , there isassigned an m-structure on S unionsq T . The analogous object in the category gVec of graded vector spaces isa graded algebra.

The species E has a monoid structure defined by sending the basis element ST to the basis elementI . For L, a monoid structure is provided by concatenation of linear orders: S,T (`1 `2) = (`1, `2).

A comonoid in the category (Sp, ) is a species c together with a morphism of species : c c c,i.e., a family of maps

S,T : c[I] c[S] c[T ],one for each ordered decomposition I = S unionsq T , which are natural, coassociative and counital.

For E, a comonoid structure is defined by sending the basis vector I to the basis vector S T . ForL, a comonoid structure is provided by restricting a total order ` on I: S,T (`) = `|S `|T .

We assume that our species q are connected, i.e., q[] = k. In this case, the (co)unital conditions for a(co)monoid force the maps S,T and S,T to be the canonical identifications if either S or T is empty.

A Hopf monoid in the category (Sp, ) is a monoid and comonoid whose two structures are compatiblein an appropriate sense, and which carries an antipode. In this paper we only consider connected Hopfmonoids. For such Hopf monoids, the existence of the antipode is guaranteed. The species E and L, withthe structures outlined above, are two important examples of (connected) Hopf monoids.

For further details on Hopf monoids in species, see Chapter 8 of [2]. The theory of Hopf monoids inspecies is developed in Part II of this reference; several examples are discussed in Chapters 12 and 13.


2.2 Lagranges theorem for connected Hopf monoidsGiven a connected Hopf monoid k in species, we let k+ denote the species defined by

k+[I] =

{k[I] if I 6= ,0 if I = .

If k is a submonoid of a monoid h, then k+h denotes the right ideal of h generated by k+. In otherwords,

(k+h)[I] =SunionsqT=IS 6=

S,T(k[S] h[T ]).

Theorem 7 Let h be a connected Hopf monoid in the category of species. If k is a Hopf submonoid of h,then h is a free left k-module. Moreover,

h = k (h/k+h)

as left k-modules (and as species).

The proof is given after a series of preparatory results. Our argument parallels Radfords proof ofTheorem 2 [16, Section 4]. The main ingredient is a result of Larson and Sweedler [7] known as thefundamental theorem of Hopf modules [11, Thm. 1.9.4]. It states that if (M,) is a left Hopf module overK, thenM is free as a leftK-module and in fact is isomorphic to the Hopf moduleKQ, whereQ is thespace of coinvariants for the coaction . Takeuchi extends this result to the context of Hopf monoids in abraided monoidal category with equalizers [19, Thm. 3.4]; a similar result (in a more restrictive setting)is given by Lyubashenko [9, Thm. 1.1]. As a special case of Takeuchis result, we have the following.

Proposition 8 Let m be a left Hopf module over a connected Hopf monoid k in species. There is anisomorphism m = k q of left Hopf modules, where

q[I] :={m m[I] | S,T (m) = 0 for S unionsq T = I , T 6= I

}.

In particular, m is free as a left k-module.

Here : m k m denotes the comodule structure, which consists of maps

S,T : m[I] k[S]m[T ],

one for each ordered decomposition I = S unionsq T .Given a comonoid h and two subspecies u,v h, their wedge is defined by

u v := 1(u h + h v).

The remaining ingredients needed for the proof are supplied by the following lemmas.

Lemma 9 Let h be a comonoid in species. If u and v are subcomonoids of h, then: (i) u v is asubcomonoid of h and u + v u v; (ii) u v = 1(u (u v) + (u v) v).


Proof: (i) Proofs of analogous statements for coalgebras, given in [1, Section 3.3], extend to this setting.(ii) From the definition, 1

(u (u v) + (u v) v) u v. Now, since u v is a subcomonoid,

(u v) ((u v) (u v)) (u h + h v) = u (u v) + (u v) v,since u,v u v. This proves the converse inclusion. 2Lemma 10 Let h be a Hopf monoid in species and k a submonoid. Let u,v h be subspecies whichare left k-submodules of h. Then u v is a left k-submodule of h.Proof: Since h is a Hopf monoid, the coproduct : h h h is a morphism of left h-modules, whereh acts on h h via . Hence it is also a morphism of left k-modules. By hypothesis, u h + h v is a leftk-submodule of h h. Hence, u v = 1(u h + h v) is a left k-submodule of h. 2Lemma 11 Let h be a Hopf monoid in species and k a Hopf submonoid. Let c be a subcomonoid of hand a left k-submodule of h. Then (k c)/c is a left k-Hopf module. 2

We are nearly ready for the proof of the main result. First, recall the coradical filtration of a connectedcomonoid in species [2, 8.10]. Given a connected comonoid c, define subspecies c(n) by

c(0) = 1 and c(n) = c(0) c(n1) for all n 1.We then have

c(0) c(1) c(n) c and c =n0

c(n).

Proof of Theorem 7: We show that there is a species q such that h = k q as left k-modules. As in theproof of Theorem 3, one then argues that q = h/k+h.

Define a sequence k(n) of subspecies of h by

k(0) = k and k(n) = k k(n1) for all n 1.Each k(n) is a subcomonoid and a left k-submodule of h. This follows from Lemmas 9 and 10, by

induction on n. Then, by Lemma 11, for all n 1 the quotient species k(n)/k(n1) is a left Hopfk-module. Therefore, by Proposition 8, each k(n)/k(n1) is a free left k-module.

We claim that there exists a sequence of species qn (n 0) such thatk(n) = k qn

as left k-modules (so that each k(n) is a free left k-module). Moreover, the qn can be chosen so that

q0 q1 qn and the above isomorphisms are compatible with the inclusions qn1 qn and k(n1) k(n). Thismay be proven by induction on n.

Finally, since h is connected, h(0) = 1 k = k(0), and by induction, h(n) k(n) for all n 0.Hence,

h =n0

h(n) =n0

k(n) =n0

k qn = k q where q =n0

qn.

Thus, h is free as a left k-module. 2


3 Applications and examples3.1 A test for Hopf submonoidsTwo important power series associated to a (finite dimensional) species q are its exponential generatingseries Eq(x) and type generating series Tq(x). They are given by

Eq(x) =n0

dim q[n]xn

n!and Tq(x) =

n0

dim q[n]Sn xn,

whereq[n]Sn = q[n]/k{v v | v q[n], Sn}.

Both are specializations of the cycle index seriesZq(x1, x2, . . . ); see [3, 1.2] for definitions. Specifically,

Eq(x) = Zq(x, 0, . . . ) and Tq(x) = Zq(x, x2, . . . ).

The cycle index series is multiplicative under Cauchy product: if h = k q, then Zh(x1, x2, . . . ) =Zk(x1, x2, . . . )Zq(x1, x2, . . . ); see [3, 1.3]. Thus, the same is true for Eq(x) and Tq(x).

Let Q0 denote the nonnegative rational numbers. A consequence of Theorem 7 is the following.

Corollary 12 Let h and k be connected Hopf monoids in species. If k is either a Hopf submonoid ora quotient Hopf monoid of h, then the quotient Zh(x1, x2, . . . )/Zk(x1, x2, . . . ) of cycle index series isnonnegative, i.e., belongs toQ0[[x1, x2, . . . ]]. Likewise for the quotients Eh(x)/Ek(x) and Th(x)/Tk(x).

Given a connected Hopf monoid h in species, we may use Corollary 12 to determine if a given speciesk may be a Hopf submonoid (or a quotient Hopf monoid).

Example 13 A partition of a set I is an unordered collection of disjoint nonempty subsets of I whoseunion is I . The notation ab.c is shorthand for the partition

{{a, b}, {c}} of {a, b, c}.Let be the species of set partitions, i.e., [I] is the vector space with basis the set of all partitions of

I . Let denote the subspecies linearly spanned by set partitions with distinct block sizes. For example,

[{a, b, c}] = k{abc, a.bc, ab.c, a.bc, a.b.c} and [{a, b, c}] = k{abc, a.bc, ab.c, a.bc}.The sequences (dim [n])n0 and (dim [n])n0 appear in [17] as A000110 and A007837, respec-tively. We have

E(x) = exp(exp(x) 1) = 1 + x+ x2 + 5

6x3 +

58x4 +

andE(x) =

n1

(1 +

xn

n!)

= 1 + x+12x2 +

23x3 +

524x4 + .

If a Hopf monoid structure on existed for which were a Hopf submonoid, then the quotient oftheir exponential generating series would be nonnegative, by Corollary 12. However, we have

E(x)/E(x) = 1 + 12x2 13x3 + 12x4 1130x5 + ,


so no such structure exists. In [2, 12.6], a Hopf monoid structure on is defined. By the above, there isno way to embed as a Hopf submonoid.

We remark that the type generating series quotient for the pair of species in Example 13 is positive:

T(x) = 1 + x+ 2x2 + 3x3 + 5x4 + 7x5 + 11x6 + 15x7 + ,T(x) = 1 + x+ x2 + 2x3 + 2x4 + 3x5 + 4x6 + 5x7 + ,

T(x)/T(x) = 1 + x2 + 2x4 + 3x6 + 5x8 + 7x10 + .

This can be understood by appealing to the Hopf algebra Sym. A basis for its nth graded piece is indexedby integer partitions, so PSym(x) = T(x). Moreover, T(x) enumerates the integer partitions with oddpart sizes and Sym does indeed contain a Hopf subalgebra with this Poincare series. It is the algebra ofSchur-Q functions. See [10, III.8]. Thus T(x)

/T(x) is nonnegative by Corollary 4.3.2 A test for Hopf monoidsLet (an)n0 be a sequence of nonnegative integers with a0 = 1. Does there exist a connected Hopfmonoid h with dim h[n] = an for all n? The next result provides conditions that the sequence (an)n0must satisfy for this to be the case.

Corollary 14 (The (ord/exp)-test) For any connected Hopf monoid in species h,(n0

dim h[n]xn)/(

n0

dim h[n]n!

xn) N[[x]].

Proof: We make use of the Hadamard product of Hopf monoids [2, Corollary 8.59]. The exponentialspecies E is the unit for this operation.

Consider the canonical morphism of Hopf monoids L E; it maps any linear order ` L[I] to thebasis element I E[I] [2, Section 8.5]. The Hadamard product then yields a morphism of Hopf monoids

L h E h = h.By Corollary 12, ELh(x)/Eh(x) N[[x]]. Since ELh(x) =

n0 dim h[n]x

n, the result follows. 2

Let an = dim h[n]. Corollary 14 states that the ratio of the ordinary to the exponential generatingfunction of the sequence (an)n0 must be nonnegative. This translates into a sequence of polynomialinequalities, the first of which are as follows:

5a3 3a2a1, 23a4 + 12a2a21 20a3a1 + 6a22.In particular, not every nonnegative sequence arises as the sequence of dimensions of a Hopf monoid.

3.3 A test for Hopf monoids over EOur next result is a necessary condition for a Hopf monoid in species to contain or surject onto theexponential species E.

Given a sequence (an)n0, its binomial transform (bn)n0 is defined by

bn :=ni=0

(n

i

)(1)i ani.


Corollary 15 (The E-test) Suppose h is a connected Hopf monoid that either contains the species E orsurjects onto E (in both cases as a Hopf monoid). Let an = dim h[n] and an = dim h[n]Sn . Then thebinomial transform of (an)n0 must be nonnegative and (an)n0 must be nondecreasing.

More plainly, in this setting, we must have the following inequalities:

a1 a0, a2 2a1 a0, a3 3a2 3a1 + a0, . . .

and an an1 for all n 1.Proof: Since EE(x) = exp(x), the quotient Eh(x)/EE(x) is given by

b0 + b1x+ b2x2

2+ b3

x3

3!+ ,

where (bn)n0 is the binomial transform of (an)n0. It is nonnegative by Corollary 12. Replacingexponential for type generating functions yields the result for (an)n0, since TE(x) = 11x . 2

We make a further remark regarding connected linearized Hopf monoids. These are Hopf monoids ofa set theoretic nature. See [2, 8.7] for details. Briefly, there are set maps A,B(x, y) and A,B(z) thatproduce single structures (written (x, y) 7 x y and z 7 (z|A, z/A), respectively), which are compatibleat the level of set maps and which produce a Hopf monoid in species when linearized. It follows that ifh is a linearized Hopf monoid, then there is a unique morphism of Hopf monoids from h onto E. Thus,Corollary 15 provides a test for existence of a linearized Hopf monoid structure on h.

Example 16 We return to the species of set partitions into distinct block sizes. We might ask if thiscan be made into a linearized Hopf monoid in some way (after Example 13, this would not be as a Hopfsubmonoid of ). With an and bn as above, we have:

(an)n0 = 1, 1, 1, 4, 5, 16, 82, 169, 541, . . . ,

(bn)n0 = 1, 0, 0, 3, 8, 25, 9, 119, 736, . . . .

Thus fails the E-test and the answer to the above question is negative.

3.4 A test for Hopf monoids over LLet h be a connected Hopf monoid in species. Let an = dim h[n] and an = dim h[n]Sn . Note that theanalogous integers for the species L of linear orders are bn = n! and bn = 1. Now suppose that h containsL or surjects onto L as a Hopf monoid. An obvious necessary condition for this situation is that an n!and an 1. Our next result provides stronger conditions.Corollary 17 (The L-test) Suppose h is a connected Hopf monoid that either contains the species L orsurjects onto L (in both cases as a Hopf monoid). If an = dim h[n] and an = dim h[n]Sn , then

an nan1 and an an1 (n 1).

Proof: It follows from Corollary 12 that both Eh(x)/EL(x) and Th(x)/TL(x) are nonnegative. Theseyield the first and second set of inequalities, respectively. 2


Before giving an application of the corollary, we introduce a new Hopf monoid in species. A composi-tion of a set I is an ordered collection of disjoint nonempty subsets of I whose union is I . The notationab|c is shorthand for the composition ({a, b}, {c}) of {a, b, c}.

Let Pal denote the species of set compositions whose sequence of block sizes is palindromic. So, forinstance,

Pal[{a, b}] = k{ab, a|b, b|a}and

Pal[{a, b, c, d, e}] = k{abcde, a|bcd|e, ab|c|de, a|b|c|d|e, . . .}.The latter space has dimension 171 = 1 + 5

(43

)+(52

)3 + 5! and dim Pal[5]S5 = 4 for the four types of

palindromic set compositions shown above.Given a palindromic set composition pi = pi1| |pir, we view it as a triple pi = (pi, pi0, pi+), where

pi is the initial sequence of blocks, pi0 is the central block if this exists (if the number of blocks is odd)and otherwise it is the empty set, and pi+ is the final sequence of blocks. That is,

pi = pi1| |pibr/2c, pi0 ={pibr/2c+1 if r is odd, if r is even, pi

+ = pidr/2+1e| |pir.

Given S I , letpi|S := pi1 S |pi2 S | |pir S ,

where empty intersections are deleted. Then pi|S is a composition of S. It is not always the case that pi|Sis palindromic. Let us say that S is admissible for pi when it is, i.e.,

#(pii S

)= #

(pir+1i S

)for all i = 1, . . . , r.

In this case, both pi|S and pi|I\S are palindromic.We now define product and coproduct operations on Pal. Fix a decomposition I = S unionsq T .

Product. Given palindromic set compositions pi Pal[S] and Pal[T ], we putS,T (pi ) :=

(pi|, pi0 0, +|pi+).

In other words, we concatenate the initial sequences of blocks of pi and in that order, merge theircentral blocks, and concatenate their final sequences in the opposite order. The result is a palindromiccomposition of I . For example, with S = {a, b} and T = {c, d, e, f},

(a|b) (c|de|f) 7 a|c|de|f |b.Coproduct. Given a palindromic set composition Pal[I], we put

S,T () :=

{ |S |T if S is admissible for pi,0 otherwise.

For example, with S and T as above,

ad|b|e|cf 7 0 and e|abcd|f 7 (ab) (e|cd|f).These operations endow Pal with the structure of Hopf monoid, as may be easily checked.


Example 18 We ask if Pal contains (or surjects onto) the Hopf monoid L. Both Hopf monoids arecocommutative and not commutative. Writing an = dim Pal[n], we have:

(an)n0 = 1, 1, 3, 7, 43, 171, 1581, 8793, 108347, . . . .

Every linear order is a palindromic set composition with singleton blocks. Thus an n! for all n and thequestion has some hope for an affirmative answer. However,

(an nan1)n1 = 0, 1, 2, 15, 44, 555, 2274, 38003, . . . ,

so Pal fails the L-test and the answer to the above question is negative.

3.5 Examples of nonnegative quotients

We comment on a few examples where the quotient power series Eh(x)/Ek(x) is not only nonnegativebut is known to have a combinatorial interpretation as a generating function.

Example 19 Consider the Hopf monoid of set partitions. It contains E as a Hopf submonoid via themap that sends I to the partition of I into singletons. We have

E(x)/EE(x) = exp(exp(x) x 1),

which is the exponential generating function for the number of set partitions into blocks of size strictlybigger than 1. This fact may also be understood with the aid of Theorem 7, as follows. The I-componentof the right ideal E+ is linearly spanned by elements of the form S pi where I = S unionsq T and pi is apartition of T . Now, since S = {i} S\{i} (for any i S), we have that E+[I] is linearly spannedby elements of the form {i} pi where i I and pi is a partition of I \ {i}. But these are precisely thepartitions with at least one singleton block.

Example 20 Let be the Hopf monoid of set compositions defined in [2, Section 12.4]. It contains L asa Hopf submonoid via the map that views a linear order as a composition into singletons. This and othermorphisms relating E, L, and , as well as other Hopf monoids, are discussed in [2, Section 12.8].

The sequence (dim [n])n0 is A000670 in [17]. We have

E(x) = 12 exp(x) .

Moreover, it is known from [18, Exercise 5.4.(a)] that

1 x2 exp(x) =

n0

snn!xn

where sn is the number of threshold graphs with vertex set [n] and no isolated vertices. Together withTheorem 7, this suggests the existence of a basis for /L+ indexed by such graphs.


References[1] Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press,

Cambridge,1980.[2] Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM

Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010.[3] Francois Bergeron, Gilbert Labelle, and Pierre Leroux, Combinatorial species and tree-like struc-

tures, Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, Cam-bridge, 1998, Translated from the 1994 French original by Margaret Readdy, with a foreword byGian-Carlo Rota.

[4] Adriano M. Garsia and Nolan Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2003),no. 2, 217263.

[5] Andre Joyal, Une theorie combinatoire des series formelles, Adv. in Math. 42 (1981), no. 1, 182.[6] Irving Kaplansky, Bialgebras, Lecture Notes in Mathematics, Department of Mathematics, Univer-

sity of Chicago, Chicago, Ill., 1975.[7] Moss E. Sweedler and Richard G. Larson, An associative orthogonal bilinear form for Hopf alge-

bras, Amer. J. Math. 91 (1969), 7594.[8] Aaron Lauve and Sarah Mason, QSym over Sym has a stable basis, available at arXiv:1003.2124v1.[9] Volodymyr Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98

(1995), no. 3, 279327.[10] Ian G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical

Monographs, Oxford University Press, New York, 1995, with contributions by A. Zelevinsky.[11] Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series

in Mathematics, vol. 82, Published for the CBMS, Washington, DC, 1993.[12] Warren D. Nichols and M. Bettina Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111

(1989), no. 2, 381385.[13] Ulrich Oberst and Hans-Jurgen Schneider, Untergruppen formeller Gruppen von endlichem Index,

J. Algebra 31 (1974), 1044.[14] Hans-Jurgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Isr. J. Math. 72

(1990), no. 12, 167195.[15] Yorck Sommerhauser, On Kaplanskys conjectures, Interactions between ring theory and represen-

tations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York,2000, pp. 393412.

[16] David E. Radford, Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45 (1977),no. 2, 266273.

[17] Neil J. A. Sloane, The on-line encyclopedia of integer sequences, published electronically atwww.research.att.com/njas/sequences/, OEIS.

[18] Richard P. Stanley, Enumerative combinatorics. Vol. 2., Cambridge Studies in Advanced Mathemat-ics, vol. 62, Cambridge University Press, Cambridge, 1999, with a foreword by Gian-Carlo Rota andappendix 1 by Sergey Fomin.

[19] Mitsuhiro Takeuchi, Finite Hopf algebras in braided tensor categories, J. Pure Appl. Algebra 138(1999), no. 1, 5982.


GelfandTsetlin Polytopes andFeiginFourierLittelmannVinberg Polytopesas Marked Poset Polytopes

Federico Ardilaand Thomas Bliemand Dido Salazar

Department of Mathematics, San Francisco State University, San Francisco, CA, USA

Abstract. Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytopeand chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. Wegeneralize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of itselements.

Through this construction, we explain combinatorially the relationship between the GelfandTsetlin polytopes (1950)and the FeiginFourierLittelmannVinberg polytopes (2010, 2005), which arise in the representation theory of thespecial linear Lie algebra.

We then use the generalized GelfandTsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjecturalanalogues of the FeiginFourierLittelmannVinberg polytopes corresponding to the symplectic and odd orthogonalLie algebras.

Resume. Stanley (1986) a montre que chaque ensemble fini partiellement ordonne permet de definir deux polye`dres,le polye`dre de lordre et le polye`dre des chanes. Ces polye`dres ont le meme polynome de Ehrhart, bien quils soienttout a` fait distincts du point de vue combinatoire. On generalise ce resultat a` une famille plus generale de polye`dres,construits a` partir dun ensemble partiellement ordonne ayant des entiers attaches a` certains de ses elements.

Par cette construction, on explique en termes combinatoires la relation entre les polye`dres de Gelfand-Tsetlin (1950)et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la theorie des representations desalge`bres de Lie lineaires speciales. On utilise les polye`dres de Gelfand-Tsetlin generalises par Berenstein et Zelevinsky(1989) afin dobtenir des analogues (conjectures) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les alge`bresde Lie symplectiques et orthogonales impaires.

Keywords: poset, polytope, semisimple Lie algebra, PBW filtration

Supported in part by the National Science Foundation CAREER Award DMS-0956178 and the National Science FoundationGrant DMS-0801075.

Supported by the Deutsche Forschungsgemeinschaft Grant SPP 1388.Supported in part by the National Science Foundation Grant DGE-0841164.


28 Federico Ardila and Thomas Bliem and Dido Salazar

1 IntroductionConsider the simple complex Lie algebra sln. The irreducible representations of sln are parametrized upto isomorphism by dominant integral weights, i.e., weakly decreasing n-tuples of integers determined upto adding multiples of (1, . . . ,1). Given a dominant integral weight , let V ( ) denote the correspond-ing irreducible sln-module. The module V ( ) has a distinguished basis, the GelfandTsetlin [5] basis,parametrized by the points with integral coordinates (integral points or lattice points for short) in theGelfandTsetlin polytope GT( ) Rn(n1)/2.

Recently, Feigin, Fourier, and Littelmann [3] constructed a different basis of V ( ), conjecturallyannounced by Vinberg [8]. This basis is related to the PoincareBirkhoffWitt basis of the universalenveloping algebra U(n), where n is the span of the negative root spaces. Again, the basis elements areparametrized by the integral points in a certain polytope FFLV( ) Rn(n1)/2.

Feigin, Fourier, and Littelmann used two subtle algebraic arguments to prove that their basis indeedspans V ( ) and is linearly independent. When they had only produced the first half of the proof, theyasked the second author of this paper:

Question 1.1. [4] Is there a combinatorial explanation for the fact that GT( ) and FFLV( ) contain thesame number of lattice points?

This question provided the motivation for this paper. We answer it by generalizing a result of Stanley [6]on poset polytopes, as we now describe. Let P be a finite poset. Let A be a subset of P which contains allminimal and maximal elements of P. Let = (a)aA be a vector in RA, which we think of as a markingof the elements of A with real numbers. We call such a triple (P,A, ) a marked poset.

Definition 1.2. The marked order polytope of (P,A, ) is

O(P,A) = {x RPA | xp xq for p< q, a xp for a< p,xp a for p< a},

where p and q represent elements of PA, and a represents an element of A. The marked chain polytopeof (P,A, ) is

C (P,A) = {x RPA0 | xp1 + + xpk bafor a< p1 < < pk < b},

where a,b represent elements of A, and p1, . . . , pk represent elements of PA.For any polytope with integer coordinates Q there exists a polynomial EQ(t), the Ehrhart polynomial

of Q, with the following property: for every positive integer n, the n-th dilate nQ of Q contains exactlyEQ(n) lattice points (see [7]). With this notion, our answer to Question 1.1 is given by the following tworesults.

Theorem 1.3. For any marked poset (P,A, ) with ZA, the marked order polytope O(P,A) and themarked chain polytope C (P,A) have the same Ehrhart polynomial.

Theorem 1.4. For every partition there exists a marked poset (P,A, ) such that GT( ) =O(P,A) andFFLV( ) = C (P,A) .

GT polytopes and FFLV polytopes as marked poset polytopes 29

We also consider the extension of these constructions to other Lie algebras. Berenstein and Zelevinskyproposed a construction of generalized GelfandTsetlin polytopes [1] for other semisimple Lie algebras.For the symplectic and odd orthogonal Lie algebras, their polytopes are also in the family of marked orderpolytopes. Therefore Theorem 1.3 yields candidates for the FeiginFourierLittelmannVinberg polytopesin types Bn and Cn.

The paper is organized as follows. In 2 we discuss the relevant aspects of the representation theoryof the simple complex Lie algebras sln. Section 3 treats marked order and chain polytopes, and gives abijection between their lattice points. Section 4 discusses the application of the combinatorial results of 3to the representation theoretic polytopes that interest us.

We note that the combinatorial 3 is self-contained, and may be of independent interest beyond therepresentation theoretic application. A possible way to read this article is to skip 2 and continue theredirectly.

2 PreliminariesConsider the simple complex Lie algebra sln. Let h be the Cartan subalgebra consisting of its diagonalmatrices. For i = 1, . . . ,n, let i h denote the projection onto the i-th diagonal component. As 1+ +n = 0, the coefficient vector of an integral weight is only determined as an element of Zn/(1, . . . ,1).We identify an integral weight with the corresponding equivalence class of coefficient vectors. If is aweight and we use the symbol in a context where it has to be interpreted as an n-tuple = (1, . . . ,n),we use the convention that a representative has been chosen implicitly. Fix simple roots i = i i+1 fori = 1, . . . ,n1. The corresponding fundamental weights are i = 1+ + i. Hence dominant integralweights correspond to weakly decreasing n-tuples of integers, or partitions.

Given a dominant integral weight , the associated GelfandTsetlin [5] polytope GT( ) is defined asfollows: Consider the board given in Figure 1.

1 2 n

Figure 1: Board defining GelfandTsetlin patterns.

Each one of the n(n1)/2 empty boxes stands for a real variable. The polytope GT( ) Rn(n1)/2is given by the fillings of the board with real numbers with the following property: each number is lessthan or equal to its upper left neighbor and greater than or equal to its upper right neighbor. Note thatthe ambiguity in choosing an n-tuple for the weight amounts to an integral translation of GT( ), andhence does not affect its number of integral points. In fact, the integral points in GT( ) parametrize theGelfandTsetlin basis of V ( ), hence |GT( )Zn(n1)/2|= dimV ( ).

Feigin, Fourier, and Littelmann [3] associate a different polytope with a dominant integral weight asfollows: The positive roots of sln are + = {i, j | 0 i < j n}, where i, j = i j. A Dyck path isby definition a sequence ( (0), . . . , (k)) in + such that (0) and (k) are simple, and if (l) = i, j,

30 Federico Ardila and Thomas Bliem and Dido Salazar

then either (l+1) = i+1, j or (l+1) = i, j+1. Denote the coordinates on R+ by s for +. Let = m11+ +mn1n1. Then the polytope FFLV( ) R+ is given by the inequalities

s 0

for all + ands (0)+ + s (k) mi+ +m j

for all Dyck paths ( (0), . . . , (k)) such that (0) = i and (k) = j.For all +, let f be a nonzero element of the root space g . Let v be a highest weight vector

of V ( ). Fix any total order on +. As s ranges over the lattice points of FFLV( ), the elements(+ f

s)v form a basis of V ( ) [3, Th. 3.11]. Hence |FFLV( )Z+ |= dimV ( ).

The previous discussion shows that |FFLV( )Z+ |= |GT( )Zn(n1)/2|. In the sequel, we give acombinatorial explanation and an extension of this fact.

3 Marked poset polytopesTo any finite poset P, Stanley [6] associated two polytopes in RP: the order polytope and the chain polytope.He showed that there is a continuous, piecewise linear bijection between them, which restricts to a bijectionbetween their sets of integral points. In this section we construct a generalization of the order and chainpolytopes, and prove the analogous result. We begin with a review of Stanleys work.

3.1 Stanleys order and chain polytopes

Let P be a finite poset. For p,q P we say that p covers q, and write p q, when p> q and there is nor P with p> r > q. We identify P with its Hasse diagram: the graph with vertex set P, having an edgegoing down from p to q whenever p covers q.

The order polytope and chain polytope of P are,

O(P) = {x [0,1]P | xp xq for all p< q}, andC (P) = {x [0,1]P | xp1 + + xpk 1 for all chains p1 < < pk}.

respectively.Stanley proved that, even though O(P) and C (P) can have quite different combinatorial structures, they

have the same Ehrhart polynomial. He did this as follows. Define the transfer map : RP RP by

(x)p =

{xp if p is minimal,min{xp xq | p q} otherwise

(1)

for x RP, p P. Then:Theorem 3.1 ([6, Theorem 3.2]). The transfer map restricts to a continuous, piecewise linear bijectionfrom O(P) onto C (P). For any m N, restricts to a bijection from O(P) 1m ZP onto C (P) 1m ZP.

GT polytopes and FFLV polytope

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