ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaTensor Fields of Mixed Young Symmetry Typeand N{ComplexesMi hel Dubois-VioletteMar Henneaux

Vienna, Preprint ESI 1088 (2001) O tober 23, 2001Supported by Federal Ministry of S ien e and Transport, AustriaAvailable via http://www.esi.a .at

TENSOR FIELDS OF MIXED YOUNG SYMMETRYTYPE AND N-COMPLEXESMi hel DUBOIS-VIOLETTE 1 and Mar HENNEAUX2De ember 5, 2001Abstra tWe onstru t N - omplexes of non ompletely antisymmetri ir-redu ible tensor �elds on RD whi h generalize the usual omplex(N = 2) of di�erential forms. Although, for N � 3, the generalized ohomology of these N - omplexes is non trivial, we prove a general-ization of the Poin ar�e lemma. To that end we use a te hnique rem-inis ent of the Green ansatz for parastatisti s. Several results whi happeared in various ontexts are shown to be parti ular ases of thisgeneralized Poin ar�e lemma. We furthermore identify the nontrivialpart of the generalized ohomology. Many of the results presentedhere were announ ed in [10℄.LPT-ORSAY 01-10ULB-TH/01-161Laboratoire de Physique Th�eorique, UMR 8627Universit�e Paris XI, Batiment 210F-91 405 Orsay Cedex, Fran eMi hel.Dubois-Violette�th.u-psud.fr2Physique Th�eorique et Math�ematiqueUniversit�e Libre de BruxellesCampus Plaine C.P. 231B-1050 Bruxelles, Belgiquehenneaux�ulb.a .be

1 Introdu tionOur aim in this paper is to develop di�erential tools for irredu ible tensor�elds on RD whi h generalize the al ulus of di�erential forms. By an irre-du ible tensor �eld on RD, we here mean, a smooth mapping x 7! T (x) ofRD into a ve tor spa e of ( ovariant) tensors of given Young symmetry. Were all that this implies that the representation of GLD in the orrespondingspa e of tensors is irredu ible.Throughout the following (x�) = (x1; : : : ; xD) denotes the anoni al o-ordinates of RD and �� are the orresponding partial derivatives whi h weidentify with the orresponding ovariant derivatives asso iated to the anon-i al at torsion-free linear onne tion (0)r of RD. Thus, for instan e, if T isa ovariant tensor �eld of degree p on RD with omponents T�1:::�p(x), then(0)r T denotes the ovariant tensor �eld of degree p + 1 with omponents��p+1T�1:::�p(x). The operator (0)r is a �rst-order di�erential operator whi hin reases by one the tensorial degree.In this ontext, the spa e (RD) of di�erential forms on RD is the gradedve tor spa e of ( ovariant) antisymmetri tensor �elds on RD with graduationindu ed by the tensorial degree whereas the exterior di�erential d is up to asign the omposition of the above (0)r with antisymmetrisation, i.e.d = (�1)pAp+1Æ (0)r: p(RD)! p+1(RD) (1)whereAp denotes the antisymmetrizer on tensors of degree p. The sign fa tor(�1)p arises be ause d a ts from the left, while we de�ned ( 0r T )�1:::�p+1 =��p+1T�1:::�p. One has d2 = 0 and the Poin ar�e lemma asserts that the oho-mology of the omplex ((RD); d) is trivial, i.e. that one has Hp((RD)) = 0,2

8p � 1 andH0((RD)) = RwhereH((RD)) = Ker(d)=Im(d) = �pHp((RD))with Hp((RD)) = Ker(d : p(RD)! p+1(RD))=d(p�1(RD)).From the point of view of Young symmetry, antisymmetri tensors or-respond to Young diagrams (partitions) des ribed by one olumn of ells, orresponding to the partition (1p), whereas Ap is the asso iated Youngsymmetrizer, (see next se tion for de�nitions and onventions).There is a relatively easy way to generalize the pair ((RD); d) whi h wenow des ribe. Let (Y ) = (Yp)p2N be a sequen e of Young diagrams su h thatthe number of ells of Yp is p, 8p 2 N (i.e. su h that Yp is a partition of theinteger p for any p). We de�ne p(Y )(RD) to be the ve tor spa e of smooth ovariant tensor �elds of degree p on RD whi h have the Young symmetrytype Yp and we let (Y )(RD) be the graded ve tor spa e �pp(Y )(RD). Wethen generalize the exterior di�erential by settingd = (�1)pYp+1Æ (0)r: p(Y )(RD)! p+1(Y ) (RD) (2)where Yp is now the Young symmetrizer on tensor of degree p asso iated tothe Young symmetry Yp. This d is again a �rst order di�erential operatorwhi h is of degree one, (i.e. it in reases the tensorial degree by one), butnow, d2 6= 0 in general. Instead, one has the following result.LEMMA 1 Let N be an integer with N � 2 and assume that (Y ) is su hthat the number of olumns of the Young diagram Yp is stri tly smaller thanN (i.e. � N � 1) for any p 2 N. Then one has dN = 0.In fa t the indi es in one olumn are antisymmetrized (see below) and dN!involves ne essarily at least two partial derivatives � in one of the olumns3

sin e there are N partial derivatives involved and at most N � 1 olumns.Thus if (Y ) satis�es the ondition of Lemma 1, the pair ((Y )(RD); d) isa N - omplex (of o hains) [19℄, [6℄, [12℄, [20℄, [7℄, i.e. here a graded ve torspa e equipped with an endomorphism d of degree 1, its N -di�erential, satis-fying dN = 0. Con erning N - omplexes, we shall use here the notations andthe results of [7℄ whi h will be re alled when needed.Noti e that p(Y )(RD) = 0 if the �rst olumn of Yp ontains more thanD ells and that therefore, if Y satis�es the ondition of Lemma 1, thenp(Y )(RD) = 0 for p > (N � 1)D.One an also de�ne a graded bilinear produ t on (Y )(RD) by setting(��)(x) = Ya+b(�(x) �(x)) (3)for � 2 a(Y )(RD), � 2 b(Y )(RD) and x 2 RD. This produ t is by onstru tionbilinear with respe t to the C1(RD)-module stru ture of (Y )(RD) (i.e. withrespe t to multipli ation by smooth fun tions). It is worth noti ing here thatone always has 0(Y )(RD) = C1(RD).In this paper we shall not stay at this level of generality; for ea h N � 2we shall hoose a maximal (Y ), denoted by (Y N ) = (Y Np )p2N, satisfying the ondition of lemma 1. The Young diagram with p ells Y Np is de�ned in thefollowing manner: write the division of p byN�1, i.e. write p = (N�1)np+rpwhere np and rp are (the unique) integers with 0 � np and 0 � rp � N � 2(np is the quotient whereas rp is the remainder), and let Y Np be the Youngdiagram with np rows of N �1 ells and the last row with rp ells (if rp 6= 0).One has Y Np = ((N � 1)np ; rp), that is we �ll the rows maximally.4

We shall denote (Y N)(RD) and p(Y N )(RD) by N (RD) and pN (RD), re-spe tively. It is lear that (2(RD); d) is the usual omplex ((RD); d) ofdi�erential forms on RD. The N - omplex (N (RD); d) will be simply de-noted by N (RD). We re all [7℄ that the (generalized) ohomology of theN - omplex N (RD) is the family of graded ve tor spa es H(k)(N (RD))k 2 f1; : : : ; N � 1g de�ned by H(k)(N (RD)) = Ker(dk)=Im(dN�k), i.e.H(k)(N (RD)) = �pHp(k)(N (RD)) withHp(k)(N(RD)) = Ker(dk : pN (RD)! p+kN (RD))=dN�k(p+k�N (RD)):The following statement is our generalization of the Poin ar�e lemma.THEOREM 1 One has H(N�1)n(k) (N(RD)) = 0, 8n � 1 and H0(k)(N (RD))is the spa e of real polynomial fun tions on RD of degree stri tly less than k(i.e. � k � 1) for k 2 f1; : : : ; N � 1g.This statement redu es to the Poin ar�e lemma for N = 2 but it is a non-trivial generalization for N � 3 in the sense that, as we shall see, the spa esHp(k)(N (RD)) are nontrivial for p 6= (N � 1)n and, in fa t, are generi ally5

in�nite dimensional for D � 3, p � N .The onne tion between the omplex of di�erential forms on RD and thetheory of lassi al gauge �eld of spin 1 is well known. Namely the sub omplex0(RD) d! 1(RD) d! 2(RD) d! 3(RD) (4)has the following interpretation in terms of spin 1 gauge �eld theory. Thespa e 0(RD)(= C1(RD)) is the spa e of in�nitesimal gauge transformations,the spa e 1(RD) is the spa e of gauge potentials (whi h are the appropriatedes ription of spin 1 gauge �elds to introdu e lo al intera tions). The sub-spa e d0(RD) of 1(RD) is the spa e of pure gauge on�gurations (whi h arephysi ally irrelevant), d1(RD) is the spa e of �eld strengths or urvaturesof gauge potentials. The identity d2 = 0 ensures that the urvatures do notsee the irrelevant pure gauge potentials whereas, at this level, the Poin ar�elemma ensures that it is only these irrelevant on�gurations whi h are for-gotten when one passes from gauge potentials to urvatures (by applying d).Finally d2 = 0 also ensures that urvatures of gauge potentials satisfy theBian hi identity, i.e. are in Ker(d : 2(RD)! 3(RD)), whereas at this levelthe Poin ar�e lemma implies that onversely the Bian hi identity hara ter-izes the elements of 2(RD) whi h are urvatures of gauge potentials.Classi al spin 2 gauge �eld theory is the linearization of Einstein geomet-ri theory. In this ase, the analog of (4) is a omplex E1 d1! E2 d2! E3 d3! E4where E1 is the spa e of ovariant ve tor �eld (x 7! X�(x)) on RD, E2 is thespa e of ovariant symmetri tensor �elds of degree 2 (x 7! h��(x)) on RD,E3 is the spa e of ovariant tensor �elds of degree 4 (x 7! R��;��(x)) on RDhaving the symmetries of the Riemann urvature tensor and where E4 is thespa e of ovariant tensor �elds of degree 5 on RD having the symmetries of6

the left-hand side of the Bian hi identity. The arrows d1; d2; d3 are given by(d1X)��(x) = ��X� (x) + ��X�(x)(d2h)��;��(x) = ����h��(x) + ����h��(x)� ����h��(x)� ����h��(x)(d3R)���;��(x) = ��R��;��(x) + ��R��;��(x) + ��R��;��(x):The symmetry of x 7! R��;��(x), � � �� � �, shows that E3 = 43(RD) andthat E4 = 53(RD); furthermore one anoni ally has E1 = 13(RD) and E2 =23(RD). One also sees that d1 and d3 are proportional to the 3-di�erential dof 3(RD), i.e. d1 � d : 13(RD)! 23(RD) and d3 � d : 43(RD)! 53(RD).The stru ture of d2 looks di�erent, it is of se ond order and in reases by2 the tensorial degree. However it is easy to see that it is proportional tod2 : 23(RD) ! 43(RD). Thus the analog of (4) is (for spin 2 gauge �eldtheory) 13(RD) d! 23(RD) d2! 43(RD) d! 53(RD) (5)and the fa t that it is a omplex follows from d3 = 0 whereas our gener-alized Poin ar�e lemma (Theorem 1) implies that it is in fa t an exa t se-quen e. Exa tness at 23(RD) is H2(2)(3(RD)) = 0 and exa tness at 43(RD)isH4(1)(3(RD)) = 0, (the exa tness at 43(RD) is the main statement of [17℄).Thus what plays the role of the omplex of di�erential forms for the spin1 (i.e. 2(RD)) is the 3- omplex 3(RD) for the spin 2. More generally, forthe spin S 2 N, this role is played by the (S + 1)- omplex S+1(RD). Inparti ular, the analog of the sequen e (4) for the spin 1 is the omplexS�1S+1(RD) d! SS+1(RD) dS! 2SS+1(RD) d! 2S+1S+1 (RD) (6)for the spin S. The fa t that (6) is a omplex was known, [4℄, it here followsfrom dS+1 = 0. One easily re ognizes that dS : SS+1(RD) ! 2SS+1(RD) is7

the generalized (linearized) urvature of [4℄. Our theorem 1 implies that se-quen e (6) is exa t: exa tness at SS+1(RD) is HS(S)(S+1(RD)) = 0 whereasexa tness at 2SS+1(RD) is H2S(1)(S+1(RD) = 0, (exa tness at SS+1(RD) wasdire tly proved in [5℄ for the ase S = 3).Finally, there is a generalization of Poin ar�e duality for N(RD), whi h isobtained by ontra tions of the olumns with the Kroneker tensor "�1:::�D ofRD, that we shall des ribe in this paper. When ombined with Theorem 1,this duality leads to another kind of results. A typi al result of this kind isthe following one. Let T �� be a symmetri ontravariant tensor �eld of degree2 on RD satisfying ��T �� = 0, (like e.g. the stress energy tensor), then thereis a ontravariant tensor �eld R���� of degree 4 with the symmetry � �� � ,(i.e. the symmetry of Riemann urvature tensor), su h thatT �� = ����R���� (7)In order to onne t this result with Theorem 1, de�ne ��1:::�D�1�1:::�D�1 =T ��"��1 :::�D�1"��1:::�D�1. Then one has � 2 2(D�1)3 (RD) and onversely, any� 2 2(D�1)3 (RD) an be expressed in this form in terms of a symmetri ontravariant 2-tensor. It is easy to verify that d� = 0 (in 3(RD)) isequivalent to ��T �� = 0. On the other hand, Theorem 1 implies thatH2(D�1)(1) (3(RD)) = 0 and therefore ��T �� = 0 implies that there is a� 2 2(D�2)3 (RD) su h that � = d2�. The latter is equivalent to (7) withR�1�2 �1�2 proportional to "�1�2:::�D"�1�2:::�D��3:::�D�3 :::�D and one veri�es that,so de�ned, R has the orre t symmetry. That symmetri tensor �elds identi- ally ful�lling ��T �� = 0 an be rewritten as in Eq. (7) has been used in [23℄and more re ently in [3℄ in the investigation of the onsistent deformationsof the free spin two gauge �eld a tion.8

Beside their usefulness for omputations (and for unifying various results)through the generalization of Poin ar�e lemma (Theorem 1) and the general-ization of the Poin ar�e duality, the N - omplexes des ribed in this paper givea lass of nontrivial examples of N - omplexes whi h are not related withsimpli ial modules. Indeed most nontrivial examples of N - omplexes onsid-ered in [6℄, [7℄, [8℄, [19℄, [21℄, [20℄ are of simpli ial type and it was shown in[7℄ that su h N - omplexes ompute the ordinary ( o)homologies of the sim-pli ial modules (see also in [20℄ for the Ho hs hild ase). Furthermore thatkind of results have been re ently extended to the y li ontext in [24℄ wherenew proofs of above results have been arried over. This does not mean thatN - omplexes asso iated with simpli ial modules are not useful; for instan ein [14℄ su h a N - omplex (related with a simpli ial Ho hs hild module) wasneeded for the onstru tion of a natural generalized BRS-theory [1℄, [18℄ forthe zero modes of the SU(2) WZNW-model, see in [9℄ for a general review.It is however very desirable to produ e useful examples whi h are not ofsimpli ial type and, apart from the universal onstru tion of [12℄ (and some�nite-dimensional examples [7℄, [12℄), the examples produ ed here are the�rst ones es aping from the simpli ial frame.Many results of this paper where announ ed in our letter [10℄ so an im-portant part of it is devoted to the proofs of these results in parti ular to theproof of Theorem 1 above whi h generalizes the Poin ar�e lemma. In orderthat the paper be self ontained we re all some basi de�nitions and resultson Young diagrams and representations of the linear group whi h are neededhere. Throughout the paper, we work in the real setting, so all ve tor spa esare on the �eld R of real numbers (this obviously generalizes to any ommu-tative �eld K of hara teristi zero). 9

The plan of the paper is the following. After this introdu tion we dis ussYoung diagrams, Young symmetry types for tensor and we de�ne in this ontext a notion of ontra tion. Se tion 3 is devoted to the onstru tion ofthe basi N - omplex of tensor �elds on RD onsidered in this paper, namelyN (RD), and the des ription of the generalized Poin ar�e (Hodge) dualityin this ontext. In Se tion 4 we introdu e a multi omplex on RD and weanalyse its ohomologi al properties; Theorem 2 proved there, whi h is byitself of interest, will be the basi ingredient in the proof of our generalizationof the Poin ar�e lemma i.e. of Theorem 1. Se tion 5 ontains this proofof Theorem 1. In Se tion 6 we analyse the stru ture of the generalized ohomology of N (RD) in the degrees whi h are not exhausted by Theorem 1.The N - omplex N(RD) is a generalization of the omplex (RD) = 2(RD)of di�erential forms on RD; in Se tion 7 we de�ne another generalization[N ℄(RD) of the omplex of di�erential forms whi h is also a N - omplex andwhi h is an asso iative graded algebra a ting on the graded spa e N(RD).In Se tion 8 whi h plays the role of a on lusion we sket h another possibleproof of Theorem 1 based on a generalization of algebrai homotopy for N - omplexes. In this se tion we also de�ne natural N - omplexes of tensor �eldson omplex manifolds whi h generalize the usual ��- omplex (of forms in d�z).2 Young diagrams and tensorsFor the Young diagrams et . we use throughout the onventions of [16℄. AYoung diagram Y is a diagram whi h onsists of a �nite number r > 0 ofrows of identi al squares (refered to as the ells) of �nite de reasing lengthsm1 � m2 � � � � � mr > 0 whi h are arranged with their left hands underone another. The lengths ~m1; : : : ; ~m of the olumns of Y are also de reasing10

~m1 � � � � � ~m > 0 and are therefore the rows of another Young diagram ~Ywith ~r = rows. The Young diagram ~Y is obtained by ipping Y over itsdiagonal (from upper left to lower right) and is refered to as the onjugateof Y . Noti e that one has ~m1 = r and therefore also m1 = ~r = and thatm1 + � � � +mr = ~m1 + � � � + ~m is the total number of ells of Y whi h willbe denoted by jY j. It is onvenient to add the empty Young diagram Y0 hara terized by jY j = 0.The �gure below des ribes a Young diagram Y andits onjugate ~Y :Y = ~Y=In the following E denotes a �nite-dimensional ve tor spa e of dimensionD and E� denotes its dual. The n-th tensor power En of E identi�es anoni ally with the spa e of multilinear forms on (E�)n. Let Y be a Youngdiagram and let us onsider that the jY j opies of E� in (E�)jY j are labelledby the ells of Y so that an element of (E�)jY j is given by spe ifying anelement of E� for ea h ell of Y . The S hur module EY is de�ned to be theve tor spa e of all multilinear forms T on (E�)jY j su h that:(i) T is ompletely antisymmetri in the entries of ea h olumnof Y , 11

(ii) omplete antisymmetrization of T in the entries of a olumnof Y and another entry of Y whi h is on the right-hand side ofthe olumn vanishes.Noti e that EY = 0 if the �rst olumn of Y has length ~m1 > D. Onehas EY � EjY j and EY is an invariant subspa e for the a tion of GL(E) onEjY j whi h is irredu ible. Furthermore ea h irredu ible subspa e of En forthe a tion of GL(E) is isomorphi to EY with the above a tion of GL(E) forsome Young diagram Y with jY j = n.Let Y be a Young diagram and let T be an arbitrary multilinear form on(E�)jY j , (T 2 EjY j). De�ne the multilinear form Y(T ) on (E�)jY j byY(T ) =Xp2RXq2C(�1)"(q)T Æ p Æ qwhere C is the group of the permutations whi h permute the entries of ea h olumn and R is the group of the permutations whi h permute the entriesof ea h row of Y . One has Y(T ) 2 EY and the endomorphism Y of EjY jsatis�es Y2 = �Y for some number � 6= 0. Thus Y = ��1Y is a proje tionof EjY j into itself, Y2 = Y, with image Im(Y) = EY . The proje tion Ywill be refered to as the Young symmetrizer (relative to E) of the Young di-agram Y . The element eY = ��1Pp2RPq2C(�1)"(q)pq of the group algebraof the group SjY j of permutation of f1; : : : ; jY jg is an idempotent whi h willbe refered to as the Young idempotent of Y .By omposition of Y as above with the anoni al multilinear mapping ofE jY j into EjY j one obtains a multilinear mapping v 7! vY of E jY j into EY .The S hur module EY together with the mapping v 7! vY are hara terizeduniquely up to an isomorphism by the following universal property: For any12

multilinear mapping � : E jY j ! F of E jY j into a ve tor spa e F satisfying(i) � is ompletely antisymmetri in the entries of ea h olumnof Y ,(ii) omplete antisymmetrization of � in the entries of a olumnof Y and another entry of Y whi h is on the right-hand side ofthe olumn vanishes,there is a unique linear mapping �Y : EY ! F su h that �(v) = �Y (vY ).By onstru tion v 7! vY satis�es the onditions (i) and (ii) above.There is an obvious notion of in lusion for Young diagrams namely Y 0 isin luded in Y , Y 0 � Y , if one has this in lusion for the orresponding subsetsof the plane whenever their upper left ells oin ide. This means for instan ethat Y 0 � Y whenever the length = m1 of the �rst row of Y is greater thanthe length 0 = m01 of the �rst row of Y 0 and that for any 1 � i � 0 thelength ~mi of the i-th olumn of Y is greater than the length ~m0i of the i-th olumn of Y 0, ( � 0 and ~mi � ~m0i for 1 � i � 0).In the following we shall need a stronger notion. A Young diagram Y 0 isstrongly in luded in another one Y and we write Y 0 �� Y if the length ofthe �rst row of Y is greater than the length of the �rst row of Y 0 and if thelength of the last olumn of Y is greater than the length of the �rst olumnof Y 0. Noti e that this relation is not re exive, one has Y �� Y if and onlyif Y is re tangular whi h means that all its olumns have the same lengthor equivalently all its rows have the same length. It is lear that Y 0 �� Yimplies Y 0 � Y . 13

Let Y and Y 0 be Young diagrams su h that Y 0 �� Y and let ~m1 � � � � �~m > 0 be the lengths of the olumns of Y and ~m01 � � � � � ~m0 0 > 0 bethe lengths of the olumns of Y 0; one has � 0 and ~m � ~m01. De�ne the ontra tion of Y by Y 0 to be the Young diagram C(Y jY 0) obtained from Yby dropping ~m01 ells of the last i.e. the -th olumn of Y , ~m02 ells of the( �1)-th olumn of Y; : : : ; ~m0 0 ells of the ( � 0+1)-th olumn of Y . If ~m isstri tly geater than ~m01 then C(Y jY 0) has olumns as Y , however if ~m = ~m01then the number of olumns of C(Y jY 0) is stri tly smaller than (it is � 1if ~m �1 is stri tly greater than ~m02, et .). Noti e that if Y is re tangular thenC(Y jY 0) �� Y and C(Y jC(Y jY 0)) = Y 0 so that Y 0 7! C(Y jY 0) is then aninvolution on the set of Young diagrams Y 0 whi h are strongly in luded in Y(Y 0 �� Y ).Let again Y and Y 0 be Young diagrams with Y 0 �� Y . Our aim is now tode�ne a bilinear mapping (T; T 0) 7! C(T jT 0) of EY �E�Y 0 into EC(Y jY 0). Thiswill be obtained by restri tion of a bilinear mapping (T; T 0) 7! C(T jT 0) ofEjY j �E�jY 0j into EjC(Y jY 0)j whi h will be an ordinary ( omplete) tensorial ontra tion. Any su h tensorial ontra tion asso iates to a ontravarianttensor T of degree jY j (i.e. T 2 EjY j) and a ovariant tensor T 0 of degreejY 0j (i.e. T 0 2 E�jY 0j) a ontravariant tensor of degree jC(Y jY 0)j, (Y 0 �� Y ).In order to spe ify su h a ontra tion, one has to spe ify the entries of T ,that is of Y , to whi h ea h entry of T 0, that is of Y 0, is ontra ted (re allingthat T is a linear ombination of anoni al images of elements of E jY j andthat T 0 is a linear ombination of anoni al images of elements of E�jY 0j). Inorder that C(T jT 0) has the right antisymmetry in the entries of ea h olumnof C(Y jY 0) when T 2 EY and T 0 2 E�Y 0, one has to ontra t the entries ofT 0 orresponding to the i-th olumn of Y 0 with entries of T orresponding to14

the ( � i+1)-th olumn of Y . The pre ise hoi e and the order of the latterentries is irrelevant up to a sign in view of the antisymmetry in the entriesof a olumn. Our hoi e is to ontra t the �rst entry of the i-th olumn ofY 0 with the last entry of the ( � i+ 1)-th olumn of Y , the se ond entry ofthe i-th olumn of Y 0 with the penultimate entry of the ( � i+1)-th olumnof Y , et . for any 1 � i � 0 (with obvious onventions). This �xes thebilinear mapping (T; T 0) 7! C(T jT 0) of EjY j � E�jY 0j into EjC(Y jY 0)j . Thefollowing �gure des ribes pi turally in a parti ular ase the onstru tion ofC(Y jY 0) as well as the pla es where the ontra tions are arried over in the orresponding onstru tion of C(T jT 0) :Y = �! �! = C(Y jY 0)Y 0 = "PROPOSITION 1 Let T be an element of EY and T 0 be an element ofE�Y 0 with Y 0 �� Y . Then C(T jT 0) is an element of EC(Y jY 0).Proof As before, we identify C(T jT 0) 2 EjC(Y jY 0)j with a multilinear formon E�jC(Y jY 0)j. To show that C(T jT 0) is in EC(Y jY 0) means verifying properties(i) and (ii) above. Property (i), i.e. antisymmetry in the olumns entries ofC(Y jY 0), is lear. Property (ii) has to be veri�ed for ea h olumn of C(Y jY 0)15

and entry on its right-hand side whi h an be hosen to be the �rst entry of a olumn on the right-hand side (in view of the olumn antisymmetry). If the olumn is the last one it has no entry on the right-hand side so there nothingto verify and if the olumn is a full olumn of Y , i.e. has not be ontra ted,whi h is the ase for the i-th olumn with i � � 0, the property (ii) followsfrom the same property for T (assumption T 2 EY ) . Thus to a hieve theproof of the proposition we only need to verify property (ii) in the ase whereboth Y and Y 0 have exa tly two olumns of lengths say ~m1 � ~m2 for Y and~m01 � ~m02 for Y 0 with ~m2 > ~m01. In this ase C(Y jY 0) has also two olumnsof lengths ~m1 � ~m02 and ~m2 � ~m01 ( ~m1 � ~m02 � ~m2 � ~m01 > 0) and one hasto verify that antisymmetrization of the �rst entry of the se ond olumn ofC(Y jY 0) with the entries of the �rst olumn (of length ~m1 � ~m02) of C(Y jY 0)in C(T jT 0) gives zero. We know that antisymmetrization with all entries ofthe �rst olumn of Y give zero (for T ); however when ontra ted with T 0this identity implies a sum of antisymmetrizations of the entries of the �rst olumn of Y 0 with the su essive entries of its se ond olumn for T 0 whi hgives zero (T 0 = E�Y 0) and redu es therefore to desired antisymmetrizationwith the ~m1 � ~m02 �rst entries.�3 Generalized omplexes of tensor �eldsThroughout this se tion (Y ) denotes not just one Young diagram but a se-quen e (Y ) = (Yp)p2N of Young diagrams Yp su h that the number of ellsof Yp is equal to p that is jYpj = p, 8p 2 N. Noti e that there is no freedomfor Y0 and Y1: Y0 must be the empty Young diagram and Y1 is the Youngdiagram with one ell. Let us denote by ^(Y )E the dire t sum �p2NEYp ofthe S hur modules EYp . This is a graded ve tor spa e with ^p(Y )E = EYp.The origin of this notation is that for the sequen e (Y 2) = (Y 2p ) of the one16

olumn Young diagrams, i.e. Y 2p is the Young diagram with p ells in one olumn for any p 2 N, then ^(Y 2)E is the exterior algebra ^E of E.In the following, we shall be interested in parti ular sequen es (Y N) =(Y Np )p2N of Young diagrams satisfying the assumption of Lemma 1 (as ex-plained in the introdu tion). The sequen e (Y N ) ontains Young diagramsY Np in whi h all the rows but the last one are of length N � 1, the last onebeing of length smaller than or equal to N � 1 in su h a way that jY Np j = p(8p 2 N). Pi turally one has for instan e for N = 5Y 53 = Y 522 = Y 524 =and so on. In this ase ^(Y N )E and ^p(Y N )E = EY Np will be simply denotedby ^NE and ^pNE respe tively. Noti e that ^pNE = 0 for p > (N � 1)D,(D = dimE), so that ^N (E) = �(N�1)Dp=0 ^pN E is �nite-dimensional.Let us assume that E is equipped with a dual volume, i.e. a non-vanishingelement " of ^DE (= ^D2 E), whi h is therefore a basis of the 1-dimensionalspa e ^D(E). It is straightforward that "(N�1) is in ^(N�1)DN E = EY N(N�1)Dbe ause (i) is obvious whereas (ii) is trivial i.e. empty. The Young dia-gram Y N(N�1)D is re tangular so that ea h Young diagram whi h is in ludedin Y N(N�1)D is in fa t strongly in luded in Y N(N�1)D; this is in parti ular the17

ase for the Y Np for p � (N � 1)D. One then de�nes a linear isomorphism� : ^NE� ! ^NE generalizing the algebrai part of the Poin ar�e (Hodge)duality by setting �! = C("(N�1)j!) (8)for ! 2 ^NE�. One has � ^pN E� = ^(N�1)D�pN E (9)for p = 0; : : : ; (N � 1)D.Let (e�)�2f1;:::;Dg be a basis of E and let (��) be the dual basis of E�. Ouraim is to be able to ompute in terms of the omponents of tensors for thevarious on epts onne ted with Young diagrams. For this, one has to de idethe linear order in whi h one writes the omponents of a tensor T 2 EjY jor, whi h is the same, of a multilinear form T on E�jY j for any given Youngdiagram Y . Sin e we have labelled the arguments (entries) of su h a T by the ells of Y and sin e the omponents are obtained by taking the argumentsamong the ��, this means that one has to hoose an order for the ells of Y(i.e a way to \read the diagram" Y ). One natural hoi e is to read the rows ofY from left to right and then from up to down (like a book); another natural hoi e is to read the olumns of Y from up to down and then from left toright. Although the �rst hoi e is very natural with respe t to the sequen es(Y N ) of Young diagrams introdu ed above and will be used later, we shall hoose the se ond way of ordering in the following. The reason is that whenT belongs to the S hur module EY , then it is (property (i)) antisymmetri inthe entries of ea h olumns. Thus if Y has olumns of lengths ~m1 � � � � � ~m (> 0 for jY j 6= 0) our hoi e is indu ed by the anoni al identi� ationEY � ^ ~m1E � � � ^ ~m E (10)18

of the S hur module EY as a subspa e of ^ ~m1E � � � ^ ~m E where ^pE =^p2E is the p-th exterior power of E. With the above hoi e, the omponents(relative to the basis (e�) of E) of T 2 EjY j read T �11:::� ~m11 ;:::;�1 :::� ~m andT 2 EY if and only if these omponents are ompletely antisymmetri in the�1r; : : : ; � ~mrr for ea h r 2 f1; : : : ; g and su h that omplete antisymmetriza-tion in the �1r; : : : ; � ~mrr and �1s gives zero for any 1 � r < s � .We have de�ned for a sequen e (Y ) = (Yp) of Young diagrams withjYpj = p (8p 2 N) the graded ve tor spa e ^(Y )E whi h an be onsideredas a generalization of the exterior algebra ^E as explained above. We nowwish to de�ne the orresponding generalization of di�erential forms. Let Mbe a D-dimensional smooth manifold. For any Young diagram Y one hasthe smooth ve tor bundle T �Y (M) over M of the S hur modules (T �x (M))Y ,x 2 M . Correspondingly, for (Y ) as above, one has the smooth bundle^(Y )T �(M) overM of graded ve tor spa es ^(Y )T �x (M). The graded C1(M)-module (Y )(M) of smooth se tions of ^(Y )T �(M) is the generalization ofdi�erential forms orresponding to (Y ). In order to generalize the exteriordi�erential one has to hoose a onne tion r on the ve tor bundle T �(M)that is a linear onne tion r on M . Su h a onne tion extends anoni allyas linear mappingsr : p(Y )(M)! p(Y )(M) C1(M)1(M)where 1(M) = 1(Y )(M) is the C1(M)-module of smooth se tions of T �(M)(i.e. of di�erential 1-forms) satisfyingr(�f) = r(�)f + � dffor any � 2 p(Y )(M) and f 2 C1(M) and where d is the ordinary di�erentialof C1(M) into 1(M). Noti e that for any sequen e (Y ) of Young diagrams19

as above, one has 0(Y ) = 0(M) = C1(M) and 1(Y )(M) = 1(M) sin e onehas no hoi e for Y0 and Y1. Let us de�ne the generalization of the ovariantexterior di�erential dr : (Y )(M)! (Y )(M) bydr = (�1)pYp+1 Æ r : p(Y )(M)! p+1(Y ) (M) (11)for any p 2 N. Noti e that dr = d on C1(M) = 0(Y )(M) and that dr isa �rst order di�erential operator. Lemma 1 in the introdu tion admits thefollowing generalization.LEMMA 2 Let N be an integer with N � 2 and assume that (Y ) is su hthat the number of olumns of the Young diagram Yp is stri tly smaller thanN for any p 2 N. Then (dr)N is a di�erential operator of order stri tlysmaller than N . If r is torsion-free, then dNr is order stri tly smaller thanN � 1. If furthermore r has vanishing torsion and urvature then one has(dr)N = 0.The proof is straightforward. In the ase N = 2, if r is torsion free, (dr)2 isnot only an operator of order zero but (dr)2 = 0 follows from the �rst Bian hiidentity; however in this ase, for (Y 2), dr oin ides with the ordinary exte-rior di�erential. For the sequen es (Y N ) = (Y Np ) we denote (Y N )(M) andp(Y N )(M) simply by N (M) and pN (M). As already mentioned 2(M) isthe graded algebra (M) of di�erential forms on M .Not everyM admits a at torsion-free linear onne tion. In the followingwe shall on entrate on N (RD) equipped with d = d(0)r where (0)r is the anoni al at torsion-free onne tion of RD. So equipped, N (RD) is a N - omplex. One has of ourse N (RD) = ^NRD� C1(RD). Let us equipRD with the dual volume " 2 ^DRD whi h is the ompletely antisymmetri ontravariant tensor of maximal degree with omponent "1:::D = 1 in the20

anoni al basis of RD. Then the orresponding isomorphism � : ^NRD� !^NRD extends by C1(RD)-linearity as an isomorphism of C1(RD)-modules,again denoted by �, of N (RD) into the spa e (of ontravariant tensor �eldson RD) ^NRD C1(RD) with�pN (RD) = ^(N�1)D�pN RD C1(RD)for any 0 � p � (N � 1)D. Let us de�ne the �rst-order di�erential operatorÆ of degree �1 on ^NRD C1(RD)Æ : ^(N�1)p+rN RD C1(RD)! ^(N�1)p+r�1N RD C1(RD)by setting ÆT = YN(N�1)p+r�1 Æ ~ÆT (12)for T 2 ^(N�1)p+rN RDC1(RD) with 0 � p < D and 1 � r � N � 1, ~Æ beingde�ned by(~ÆT )�11:::�p+11 ;:::;�1r�1:::�p+1r�1;�1r :::�pr ;:::;�1N�1:::�pN�1 = ��T �11:::�p+11 ;:::;�1r :::�pr�;:::;�1N�1:::�pN�1where we have used the anoni al identi� ation (10) and the onventionsexplained below (10). It is worth noti ing here that in view (essentially) ofProposition 1, one has ÆT = ~ÆT for r = N � 1, i.e. in this ase (well-�lled ase) the proje tion is not ne essary in formula (12). So de�ned (ÆT )(x)is by onstru tion in ^(N�1)p+r�1N RD and the operator Æ is in ea h degreeproportional to the operator �d��1, i.e. that one hasÆ = n � d ��1 : ^nNRD C1(RD)! ^n�1N RD C1(RD) (13)for some n 2 R, 1 � n � (N � 1)D (Æ = 0 in degree zero).21

4 Digression on a related multi omplexIn this se tion, we introdu e a multi omplex whi h will be related to ourN - omplex N (RD) in the next se tion. We also derive some useful oho-mologi al results in this multi omplex, whi h will be the key for proving ourgeneralization of the Poin ar�e lemma that is Theorem 1.Let A be the graded tensor produ t of N�1 opies of the exterior algebra^RD� of the dual spa e RD� of RD with C1(RD),A = (N�1 ^RD�) C1(RD) = N�1C1(RD)(RD):An element of A is as a sum of produ ts of the (N � 1)D generators dix�(i = 1; : : : ; N � 1, � = 1; : : : ;D) with smooth fun tions on RD. Elementsof A will be refered to as multiforms. The spa e A is a graded- ommutativealgebra for the total degree, in parti ular one hasdix� djx� = �djx� dix�; x� dix� = dix� x�:One de�nes N � 1 antiderivations di on A by settingdif = ��f dix� (f 2 C1(RD)) ; di(djx�) = 0: (14)These antiderivations anti ommute,didj + djdi = 0 (15)in parti ular ea h di is a di�erential. The graded algebra A has a naturalmultidegree (d1; d2; : : : ; dN�1) for whi h di(djx�) = Æij.It is useful to onsider the subspa es A(k) of multiforms that vanish atthe origin, together with all their su essive derivatives up to order k � 1in luded (k � 1). If ! 2 A(k), one says that ! is of order k. The terminology22

omes from the fa t that a smooth fun tion belongs to A(k) if and only if itslimited Taylor expansion starts with terms of order � k. If l � k, A(l) � A(k).The subspa es A(k) are not stable under di but one has diA(k) � A(k�1) fork � 1 (with A(0) � A). The ve tor spa e H(k)(di;A) is de�ned asH(k)(di;A) � Z(k)(di;A)diA(k+1)where Z(k)(di;A) is the set of di- o y les 2 A(k). Note that any multiform! 2 A an be written as ! = p(k) + � where p(k) is a polynomial multiformof polynomial degree k and � 2 A(k+1). This de omposition is unique whi himplies in parti ular that A(k) \ diA = diA(k+1).It follows from the standard Poin ar�e lemma thatH(1)(di;A) = 0: (16)Indeed, the ohomology of di in A is isomorphi to the spa e of onstantmultiforms not involving dix�. The ondition that the o y les belong toA(1), i.e., vanish at the origin, eliminates pre isely the onstants. One hasalso H(m)(di;A) = 0 8m � 1 sin e A(m) � A(1) for m � 1 and A(m) \ diA =diA(m+1).Let K be an arbitrary subset of f1; 2; : : : ; N � 1g. We de�ne AK as thequotient spa e AK = APj2K djA(for K = ;, AK = A). The di�erential di indu es, for ea h i, a well-de�neddi�erential in AK whi h we still denote by di. Of ourse, the indu ed di isequal to zero if i 2 K.LEMMA 3 For every proper subset K of f1; 2; : : : ; N � 1g and for everyi =2 K, one has H(k+1)(di;AK) = 0 (k = #K)23

Proof The proof pro eeds by indu tion on the number k of elements of K.The lemma learly holds for k = 0 (K = ;) sin e then AK = A and thelemma redu es to Eq. (16). Let us now assume that the lemma holds forall subsets K (not ontaining i) with k � ` elements. Let K 0 be a subsetnot ontaining i with ` + 1 elements. Let j 2 K 0 and K 00 = K 0nfjg. Theindu tion hypothesis implies H(`+1)(di;AK00) = H(`+1)(dj ;AK00) = 0. Bystandard \des ent equation" arguments (see below), this leads toHp;q;(`+2)(dijdj;AK00) ' Hp+1;q�1;(`+2)(dijdj;AK00):In Hp;q;(`+2)(dijdj;AK00), the �rst supers ript p stands for the di-degree, these ond super ript q stands for the dj-degree while (` + 2) is the polynomialorder. Repeated appli ation of this isomorphism yieldsHp;q;(`+2)(dijdj ;AK00) ' Hp+q;0;(`+2)(dijdj ;AK00):But Hp+q;0;(`+2)(dijdj ;AK00) � Hp+q;0;(`+2)(di;AK00) = 0. Hen e, the ohomo-logi al spa es Hp;q;(`+2)(dijdj ;AK00) vanish for all p, q, whi h is pre isely thestatement H(`+2)(di;AK0) = 0.�The pre ise des ent equation argument involved in this proof runs asfollows: let �p;q;(`+2) be a di- o y le modulo dj in AK00, i.e., a solutionof di�p;q;(`+2) + dj�p+1;q�1;(`+2) � 0 for some �p+1;q�1;(`+2), where the no-tation � means \modulo terms in Pj2K00 djA. Applying di to this equa-tion yields djdi�p+1;q�1;(`+2) � 0 and hen e, sin e di�p+1;q�1;(`+2) is of order`+ 1 and H(`+1)(dj ;AK00) = 0, di�p+1;q�1;(`+2) + dj�p+2;q�2;(`+2) � 0 for some�p+2;q�2;(`+2). Hen e, �p+1;q�1;(`+2) is also a di- o y le modulo dj in AK00.Consider the map �p;q;(`+2) 7! �p+1;q�1;(`+2) of di- o y les modulo dj . There isan arbitrariness in the hoi e of �p+1;q�1;(`+2) given �p;q;(`+2) so this map is am-biguous, however H(`+1)(dj;AK00) = 0 implies that it indu es a well-de�ned24

linear mappingHp;q;(`+2)(dijdj ;AK00)! Hp+1;q�1;(`+2)(dijdj ;AK00) in ohomol-ogy. This map is inje tive and surje tive sin e H(`+1)(di;AK00) = 0 and thusone has the isomorphism Hp;q;(`+2)(dijdj ;AK00) ' Hp+1;q�1;(`+2)(dj jdi;AK00)(see [11℄ for additional information).A dire t appli ation of this lemma is the followingPROPOSITION 2 Let J be any non-empty subset of f1; 2; : : : ; N � 1g.Then �Yj2J dj�� = 0 and � 2 A(#J) ) � =Xj2J dj�jfor some �j's.Proof The property is learly true for #J = 1 (see Eq. (16)). Assumethen that the property is true for all proper subsets with k � ` < N � 1elements. Let J be a proper subset with exa tly ` elements and i =2 J . Let� be a multiform in A(`+1) su h that di(Qj2J dj)� = 0. This is equivalentto (Qj2J dj)di� = 0. Appli ation of the re ursive assumption to di�, whi hbelongs to A(`), implies then di� =Pj2J dj�j, from whi h one derives, usingthe previous lemma, that � = di� +Pj2J �j for some �, �j. Therefore, theproperty passes on to all subsets with ` + 1 elements, whi h establishes thetheorem.�We are now in position to state and prove the main result of this se tion.THEOREM 2 Let K be an arbitrary non-empty subset of f1; 2; : : : ; N�1g.If the multiform ! is su h that�Yi2I di�! = 0 8I � K j#I = m (17)25

(with m � #K a �xed integer), then! = XJ � K#J = #K �m+ 1�Yj2J dj��J + !0 (18)where !0 is a polynomial multiform of degree � m� 1.Proof The polynomial multiform !0 is learly a solution of the problem,so we only need to he k that if ! 2 A(m) in addition to (17), then (18) isrepla ed by ! = XJ � K#J = #K �m+ 1�Yj2J dj��J : (19)The �J 's an be assumed to be of order #K + 1 sin e one di�erentiatesthem #K � m + 1 times to get !. To prove (19), we pro eed re ursively,keeping m �xed and in reasing the size of K step by step from #K = mto #K = N � 1. If #K = m, there is nothing to be proven sin e I = Kand the theorem redu es to the previous theorem. So, let us assume that thetheorem has been proven for #K = k � m and let us show that it extendsto any set U = K [ f`g, ` =2 K with #U = k + 1 elements. If (17) holds forany subset I � U of U (with #I = m), it also holds for any subset I � K ofK � U (with #I = m), so the re ursive hypothesis implies! = XJ � K#J = k �m+ 1�Yj2J dj��J : (20)Let now A be an arbitrary subset of U with #A = m, whi h ontains theadded element `. Among all the subsets J o urring in the sum (20), there26

is only one, namely J 0 = UnA su h that J 0\A = ;. The ondition (17) withI = A implies, when applied to the expression (20) of !,�Yj2U dj��J 0 = 0(if J 6= J 0, the produ t (Qi2A di)(Qj2J dj) identi ally vanishes be ause atleast one di�erential df is repeated). But sin e �J 0 is of order k + 1 = #U ,the previous proposition implies that�J 0 =Xj2U dj�J 0j :When inje ted into (20), this yields in turn! = XL � U#L = k �m+ 2�Yj2L dj��0L: (21)for some �0L, and shows that the required property is also valid for sets with ardinal equal to k + 1, ompleting the proof of the theorem.�5 The generalization of the Poin ar�e lemmaWith the result of last se tion, Theorem 2, we an now pro eed to the proofof Theorem 1 that is to the proof of the generalization of the Poin ar�e lemmaannoun ed in the introdu tion.Let us �rst show that N (RD) identi�es anoni ally as graded C1(RD)-module with the image of a C1(RD)-linear homogeneous proje tion � of Ainto itself: N (RD) = �(A) � A. Indeed by using the anoni al identi� ation(10) of Se tion 3, one has the identi� ation^(N�1)n+iN E � ^n+1E � � � ^n+1E| {z }i ^nE � � � ^nE| {z }N�1�i (22)27

of the S hur module EY N(N�1)n+i = ^(n+1)n+iN E as ve tor subspa e of the right-hand side. However by de omposing the right-hand side of (22) into irre-du ible subspa es for the a tion of GL(E) one sees that there is only oneirredu ible fa tor isomorphi to EY N(N�1)n+i whi h is therefore the image of aGL(E)-invariant proje tion. It follows that ^NE � N�1 ^ E is the imageof a GL(E)-invariant proje tion P of N�1 ^ E into itself whi h is homo-geneous for the total degree. The result for N (RD) follows by hosing Eto be the dual spa e RD� of RD and by setting � = P IC1(RD) in viewof N (RD) = ^NRD� C1(RD) and A = (N�1 ^ RD�) C1(RD). Theproje tion � is in fa t by onstru tion a proje tion of �p2NA[p℄ into itselfwhere A[p℄ = An+1;:::;n+1;n;:::;n, p = (N � 1)n + i with obvious notations.We now relate the N -di�erential d of N (RD) to the di�erentials di of A.Let ! be an element of pN (RD) with p = (N � 1)n+ i, 0 � i < N � 1. Onehas d! = !�(di+1!) (23)where ! is a non-vanishing number that depends on the degrees of !. Ingeneral, the proje tion is non trivial, in the sense that di+1! has omponentsnot only along the irredu ible S hur module EY Np+1 (E = RD�), but also alongother S hur modules not o urring in N (RD). For instan e, with N = 3,the ovariant ve tor with omponents v� de�nes the element v = v�d1x�of A. One has d2v = ���v�d1x�d2x�. This expression ontains both asymmetri (dv) and an antisymmetri part, so d2v = dv � v[�;�℄d1x�d2x�.The proje tion removes v[�;�℄d1x�d2x�, whi h does not vanish in general.Be ause the proje tion is non-trivial, the onditions d! = 0 and di+1! = 0are inequivalent for generi i. However, if ! is a well-�lled tensor that is if28

i = 0, then d! = d1! (i = 0) (24)Indeed, d1! has automati ally the orre t Young symmetry. Thus the on-ditions d1! = 0 and d! = 0 are equivalent. Furthermore, be ause of thesymmetry between the olumns, if d1! = 0, then, one has also d2! =d3! = � � � = 0. For instan e, again for N = 3, the derivative of thesymmetri tensor g = g��d1x�d2x� (g�� = g��) is given by dg = d1g =12(g��;� � g��;�)d1x�d1x�d2x�. The ompletely symmetri omponent g(��;�)is absent be ause d1x�d1x� = �d1x�d1x�. Also, it is lear that if d1g = 0,then, d2g = 12(g��;�� g��;�)d1x�d2x�d2x� = 0. This generalizes to the follow-ing lemma:LEMMA 4 Let ! 2 (N�1)nN (RD) (well-�lled, or re tangular, tensor). Thendk! = 0 , ( Yj2J;#J=k dj)! = 0: (25)Proof One has dk! = (�1)md1d2 � � � dk!. Indeed, it is lear that the mul-tiform d1d2 � � � dk! 2 An+1;n+1;��� ;n+1;n;��� ;n belongs to N(RD) be ause it an-not have omponents along the invariant subspa es orresponding to Youngdiagrams with �rst olumn having i > r + 1 boxes, sin e one annot puttwo derivatives ��, �� in the same olumn. Hen e, dk! = 0 is equivalentto d1d2 � � � dk! = 0. One ompletes the proof by observing that for well-�lled tensors, the ondition d1d2 � � � dk! = 0 is equivalent to the onditionsdi1di2 � � � dik! = 0 8(i1; � � � ; ik) be ause of symmetry in the olumns. �LEMMA 5 Let ! 2 (N�1)nN (RD) with n � 1. Then! = XJ;#J=N�k �Yj2J dj��J ) ! = dN�k� (26)29

for some � 2 (N�1)n�N+kN (RD), k 2 f1; : : : ; N � 1g.Proof First, we note that the �J o urring in (26) an be hosen to havedi-degrees equal to n � 1 or n a ording to whether di a ts or does not a ton �J , sin e ! has multidegree (n; n; � � � ; n). Se ond, one an proje t theright-hand side of (26) on (N�1)nN (RD) without hanging the left-hand side,sin e ! 2 (N�1)nN (RD). It is easy to see that �[(Qj2J dj)�J ℄ � dN�k�0J ,with �0J = �(~�J), where ~�J is the element in An;��� ;n;n�1;n�1;��� ;n�1 obtainedby reordering the \ olumns" of �J so that they have non-in reasing length.In fa t, when di�erentiated, the other irredu ible omponents of ~�J do not ontribute to ! be ause their �rst olumn is too long to start with or be- ause two partial derivatives �nd themselves in the same olumn, yieldingzero. Inje ting the above expression for �[(Qj2J dj)�J ℄ into (26) yields thedesired result. �LEMMA 6 Let ! 2 (N�1)nN (RD) with n � 1 be a polynomial multiform ofdegree k � 1. Then, ! = dN�k� (27)for some polynomial multiform � 2 (N�1)n�N+kN (RD) of degree N � 1, withk 2 f1; : : : ; N � 1g.Proof The proof amounts to play with Young diagrams. The oeÆ ientsof ! transform in the tensor produ t of the representation asso iated withY N(N�1)n (symmetry of !) and the ompletely symmetri representation withk�1 boxes (symmetri polynomials in the x�'s of degree k�1). Let T be thisrepresentation and VT be the arrier ve tor spa e. Similarly, the multiform�transforms (if it exists) in the tensor produ t of the representation asso iatedwith Y N(N�1)n�N+k (symmetry of �) and the ompletely symmetri represen-30

tation with N�1 boxes (symmetri polynomials in the x�'s of degree N�1).Let S be this representation and WS be the arrier ve tor spa e. Now, thelinear operator dN�k : WS ! VT is an intertwiner for the representations Sand T . To analyse how it a ts, it is onvenient to de ompose both S and Tinto irredu ible representations.The ru ial fa t is that all irredu ible representations o urring in T alsoo ur in S. That is, if T = �iTi; VT = �iVi(where ea h irredu ible representation Ti has multipli ity one), thenS = (�iTi)� (��T�); WS = (�iWi)� (��W�)where T� are some other representations, irrelevant for our purposes. Be- ause Ti is irredu ible, the operator dN�k maps the invariant subspa e Wi onthe invariant subspa e Vi, and furthermore, dN�kjWi is either zero or bije -tive. It is easy to verify by taking simple examples that dN�kjWi is not zero.Hen e, dN�kjWi is inje tive, whi h implies that dN�k :WS ! VT is surje tive,so that ! an indeed be written as dN�k� for some �. �Proof of Theorem 1 The theorem 1 is a dire t onsequen e of the abovetwo lemmas. (i) Let ! 2 (N�1)nN (RD) (with n � 1) be annihilated by dk,dk! = 0. We write ! = !0 + !0, where !0 is of order k and where !0 isa polynomial multiform of polynomial degree < k. Both !0 and !0 havethe symmetry of !. Also, sin e !0 is trivially annihilated by dk, one hasseparately dk!0 = 0 and dk!0 = 0. We onsider �rst !0. The �rst lemmaimplies (Qj2J;#J=k dj)!0 = 0, from whi h it follows, using the theorem ofthe previous se tion, that!0 = XJ;#J=N�k �Yj2J dj��J31

(see (19)). By the se ond lemma above, this term an be written as dN�k�.As we have also seen, the same property holds for !0. This proves the the-orem for n � 1. (ii) The ase of H0(k)(N (RD)) is even easier to dis uss: fora fun tion, the ondition dkf = 0 is equivalent to ��1����kf = 0 and thus, fmust be of degree stri tly less than k. Moreover, it an never be the dN�k ofsomething, sin e there is nothing in negative degree. �It is worth noti ing here that, as explained in the introdu tion, Theorem 1has a dual ounterpart for the Æ-operator introdu ed at the end of Se tion 3whi h allows to integrate lots of generalized urrents onservation equations.In the last se tion of this paper we shall sket h another approa h for provingTheorem 1 whi h is based on the appropriate generalization of homotopy forN - omplexes.6 Stru ture of Hm(k)(N(RD)) for generi mIn the previous se tion we have shown that Hm(k)(N (RD)) vanishes wheneverm = (N�1)n with n � 1. In the ase N = 2 this is the usual Poin ar�e lemmawhi h means that the ohomology vanishes in positive degrees. For N � 3there are degrees m whi h do not belong to the set f(N � 1)(n + 1)jn 2 Ngand it turns out that for su h a (generi ) degree m, the spa es Hm(k)(N (RD))are non trivial (k 2 f1; : : : ; N � 1g). More pre isely for m 2 f0; : : : ; N � 2gthese spa es are �nite-dimensional of stri tly positive dimensions whereasfor m � N with m 6= (N � 1)n these spa es are in�nite-dimensional. Inthe following we shall expli itly display the ase N = 3 and indi ate how topro eed for the general ase N � 3.In order to simplify the notations let us denote the spa es Hm(k)(N (RD))32

by Hm(k) and the graded spa es H(k)(N(RD)) by H(k)(= �mHm(k)).For N = 3, one has only H(1) and H(2) and Theorem 1 states that H2n(1) =H2n(2) = 0 for n � 1 and that H0(1) ' R is the spa e of onstant fun tionson RD whereas H0(2) is the spa e of polynomial fun tions of degree less orequal to one on RD, i.e. H0(1) ' R� RD. On the other hand, the elementsof H1(1) identify with the ovariant ve tor �elds (or one-forms) x 7! X(x) onRD satisfying ��X� + ��X� = 0 (28)whi h is the equation hara terizing the Killing ve tor �elds (i.e. in�nites-imal isometries) of the standard eu lidean metri PD�=0(dx�)2 of RD. Thegeneral solution of (28) is X� = v�+ a��x� with v 2 RD (in�nitesimal trans-lations) and a 2 ^2RD i.e. a�� = �a��=Cte (in�nitesimal rotations). Thusone has H1(1) ' RD � ^2RD. Noti e that with this terminology we haveimpli itly identi�ed ovariant ve tor �elds with ontravariant ones by usingthe standard metri of RD. Noti e also that as far as H0(1), H0(2) and H1(1)are on erned nothing hange if N � 3. For N = 3, H1(2) identi�es with thespa e of ovariant ve tor �elds x 7! X(x) on RD satisfying��(��X� � ��X�) = 0 (29)modulo the ones of the form X� = ��' for some ' 2 C1(RD). The generalsolution of (29) is X� = a��x� + ��' with a 2 ^2RD and ' 2 C1(RD) sothat one has H1(2) ' ^2RD. Let us now show that H3(1) is in�nite-dimensionalfor N = 3. For this, onsider an arbitrary 2-form ! i.e. an arbitrary ovari-ant antisymmetri tensor �eld of degree 2 on RD and onsider the elementt = Y33Æ (0)r ! of 33(RD). Up to an irrelevant normalization onstant, the33

omponents of t are given byt��� = 2��!�� + ��!�� � ��!�� (30)and one veri�es that one has dt = 0 in 3(RD). On the other hand one hast = dh in 3(RD) that is2��!�� + ��!�� � ��!�� = ��h�� � ��h�� (31)for some symmetri ovariant tensor �eld h 2 23(RD) if and only if ! is ofthe form !�� = a���x� + ��X� � ��X� (32)for a 2 ^3RD and some ovariant ve tor �eld X 2 13(RD) and then t isproportional to d2(X) in 3(RD) i.e. t is trivial in H3(1). This argumentshows �rstly that H3(1) ontains the quotient of the spa e of 2-forms by theones of the form given by (32) whi h is in�nite-dimensional and se ondlythat the same spa e identi�es with a subspa e of H3(2) whi h is therefore alsoin�nite-dimensional. In fa t as will be shown below one has an isomorphismH3(1) ' H3(2) whi h is indu ed by the in lusion Ker(d) � Ker(d2). By repla -ing the 2-form ! by an irredu ible ovariant tensor �eld !n of degree 2n+ 2on RD with Young symmetry type given by the Young diagram with n linesof length two and two lines of length one, it an be shown similarily thatH2(n+1)+1(1) and H2(n+1)+1(2) are in�nite-dimensional spa es (we shall see thatthey are in fa t isomorphi ).The last argument for N = 3 admits the following generalization forN � 3. Let Y Nm be a Young diagram of the sequen e (Y N ) and let Y 0m�p be aYoung diagram obtained by deleting p boxes of Y Nm with 0 < p < N � 1 su hthat it does not belong to (Y N) (i.e. Y 0m�p 6= Y Nm�p) and su h that by applying34

p derivatives (i.e. (0)rp) to a generi tensor �eld with Young symmetry Y 0m�pone obtains a tensor whi h has a nontrivial omponent t with Young symme-try Y Nm . Then generi ally the latter t 2 mN (RD) is a nontrivial generalized o y le and one obtains by this pro edure an in�nite dimensional subspa eof the orresponding generalized ohomology, i.e. of Hm(k) for the appropriatek. Noti e that this is only possible for m � N with m 6= (N � 1)n. We onje ture that the whole nontrivial part of the generalized ohomology ofN (RD) in degree m � N is obtained by the above onstru tion (N � 3).In order to omplete the dis ussion for N � 3 in degree m � N � 2as well as to show the isomorphisms H2n+1(1) ' H2n+1(2) for N = 3, n � 1and their generalizations for N � 3, we now re all a basi lemma of thegeneral theory of N - omplexes [7℄, [12℄. This lemma was formulated in [7℄in the more general framework of N -di�erential modules (Lemma 1 of [7℄)that is of k-modules equipped with an endomorphism d su h that dN = 0where k is a unital ommutative ring. In this paper we only dis uss N - omplexes of (real) ve tor spa es. Let E be a N - omplex of o hain [7℄like N (RD), that is here E = �m2NEm is a graded ve tor spa e equippedwith an endomorphism d of degree one su h that dN = 0 (N � 2). Thein lusions Ker(dk) � Ker(dk+1) and Im(dN�k) � Im(dN�k�1) indu e linearmappings [i℄ : H(k) ! H(k+1) in generalized ohomology for k su h that1 � k � N � 2. Similarily the linear mappings d : Ker(dk+1)! Ker(dk) andd : Im(dN�k�1) ! Im(dN�k) obtained by restri tion of the N -di�erential dindu e linear mappings [d℄ : H(k+1) ! H(k). One has the following lemma(for a proof we refer to [12℄ or [7℄).LEMMA 7 Let the integers k and ` be su h that 1 � k, 1 � `, k+` � N�1.35

Then the hexagon of linear mappingsH(`+k)(E) H(`)(E)H(k)(E) H(N�k)(E)H(N�`)(E) H(N�(`+k))(E)-[d℄k HHHHHHj[i℄N�(`+k)������*[i℄` ������� [d℄`HHHHHHY[d℄N�(`+k) �[i℄kis exa t.Sin e [i℄ is of degree zero while [d℄ is of degree one, these hexagons give longexa t sequen es.Let us apply the above result to the N - omplex N (RD). For N = 3,there is only one hexagon as above (k = ` = 1) and, by using H2n(k) = 0 forn � 1, k = 1; 2 it redu es to the exa t sequen es0 [d℄! H0(1) [i℄! H0(2) [d℄! H1(1) [i℄! H1(2) d! 0 (33)and 0 d! H2n+1(1) [i℄! H2n+1(2) d! 0 (34)for n � 1. The sequen es (34) give the announ ed isomorphisms H2n+1(1) 'H2n+1(2) while the 4-terms sequen e (33) allows to ompute the �nite dimensionof H1(2) knowing the one of H0(1), H0(2) and H1(1). For N � 3 one has severalhexagons and by using H(N�1)n(k) = 0 for n � 0, the sequen e (33) generalizesas the following (N�2)(N�1)2 four-terms exa t sequen es0 [d℄k�! Hk�1(`) [i℄N�k�`�! Hk�1(N�k) [d℄`�! Hk+`�1(N�k�`) [i℄k�! Hk+`�1(N�`) [d℄N�k�`�! 0 (35)for 1 � k; ` and k + ` � N � 1. There are also two-terms exa t sequen esgeneralizing (34) giving similar isomorphisms but, for N > 3, there are other36

longer exa t sequen es (whi h are of �nite lengths in view of H(N�1)n(k) = 0 forn � 1). Suppose that the spa esHm(k) are �nite-dimensional for k+m � N�1and that we know their dimensions. Then the exa t sequen es (35) implythat all the Hm(k) for m � N �2 are �nite-dimensional and allows to omputetheir dimensions in terms of the dimensions of the Hm(k) for k +m � N � 1.To omplete the dis ussion it thus remains to show the �nite-dimensionalityof the Hm(k) for k +m � N � 1. For k +m � N � 1, the spa e Hm(k) identi�eswith the spa e of ( ovariant) symmetri tensor �elds S of degree m on RDsu h that X�2Sk+m ���(1) : : : ���(k)S��(k+1) : : : ��(k+m) = 0 (36)for �i 2 f1; : : : ;Dg where Sk+m is the group of permutation of f1; : : : ; k+mg.In parti ular, for k = 1 the equation (36) means that S is a Killing tensor�eld of degree m for the anoni al metri of RD and it is well known andeasy to show that the omponents of su h a Killing tensor �eld of degree mare polynomial fun tions on RD of degree less or equal to m. In fa t theKilling tensor �elds on RD form an algebra for the symmetri produ t overea h point of RD whi h is generated by the Killing ve tor �elds (whi h arepolynomial of degree � 1). Thus Hm(1) is �nite-dimensional for 1+m � N�1.By using this together with Theorem 1, one shows by indu tion on k thatHm(k) is �nite-dimensional for k+m � N�1, more pre isely, that the solutionsof (36) are polynomial fun tions on RD of degree less than k +m.The results of this se tion on erning the generi degrees show that ourgeneralization of the Poin ar�e lemma, i.e. Theorem 1, is far from being astraightforward result and that it is optimal.37

7 AlgebrasLet E ' RD be a D-dimensional ve tor spa e, (Y ) be a sequen e (Y ) =(Yp)p2N of Young diagrams su h that jYpj = p (8p 2 N) and let us use thenotations and onventions of Se tion 3. As we have seen, the graded spa e^(Y )E is a generalization of the exterior algebra of E in the sense that asgraded ve tor spa e it redu es to the latter when (Y ) oin ides with thesequen e (Y 2) = (Y 2p )p2N of the one- olumn Young diagrams. It is also ageneralization of the symmetri algebra of E sin e it redu es to it when (Y ) oin ides with the sequen es ( ~Y 2) = (~Y 2p )p2N of the one-line Young diagrams(whi h are the onjugates of the Y 2p ). In fa t, for general (Y ) the gradedve tor spa e ^(Y )E is also a graded algebra if one de�nes the produ t bysetting TT 0 = Yp+p0(T T 0) (37)for T 2 EYp and T 0 2 EYp0 where Yn is the Young symmetrizer de�nedin Se tion 2. However, although it generalizes the exterior produ t, thisprodu t is generi ally a nonasso iative one. Thus ^(Y )E is a generalizationof the exterior algebra ^E in whi h ea h homogeneous subspa e is irredu iblefor the a tion of GL(E) ' GLD but in whi h one loses the asso iativity ofthe produ t. There is another losely related generalization of the exterioralgebra onne ted with the sequen e (Y ) in whi h what is retained is theasso iativity of the graded produ t but in whi h one generi ally loses theGL(E)-irredu ibility of the homogeneous omponents. This generalization,denoted by ^[(Y )℄E, is su h that ^(Y )E is a graded (right) ^[(Y )℄E-module.We now des ribe its onstru tion.Let T(E) be the tensor algebra of E, we use the produ t de�ned by (37)38

to equip ^(Y )E with a right T(E)-module stru ture by settingT�(Y )(X1 � � � Xn) = (� � � (TX1) � � � )Xn (38)for any Xi 2 E and T 2 ^(Y )E. By de�nition the kernel Ker(�(Y )) of �(Y )is a two-sided ideal of T(E) so that the right a tion of T(E) on ^(Y )E is infa t an a tion of the quotient algebra ^[(Y )℄E = T(E)=Ker(�(Y )). So ^(Y )Eis a graded right ^[(Y )℄E-module.LEMMA 8 Let N be an integer with N � 2 and assume that (Y ) is su hthat the number of olumns of the Young diagram Yp is stri tly smaller thanN for any p 2 N. Then Ker(�(Y )) ontains the two-sided ideal of T(E) whi h onsists of the tensors whi h are symmetri with respe t to at least N of theirentries; in parti ular (�(Y )(X))N = 0, 8X 2 E.Stated di�erently, under the assumption of the lemma for (Y ), a monomialX1 : : :Xn 2 ^[(Y )℄E with Xi 2 E vanishes whenever it ontains N times thesame argument, that is if there areN distin t elements i1; : : : ; iN of f1; : : : ; ngsu h that Xi1 = � � � = XiN .Proof This is straightforward, as for the proof of Lemma 1, sin e one hasmore than N symmetrized entries whi h are distributed among less thanN � 1 olumns in whi h the entries are antisymmetrized.�The right a tion �(Y N) of T(E) on ^NE will also be simply denoted by�N . In the ase N = 2, ^2E is the usual exterior algebra ^E of E andthe right a tion �2 of T(E) fa torizes through the right a tion of ^E onitself, in parti ular Ker(�2) is the two-sided ideal of T(E) generated by theX X for X 2 E. Thus the graded algebra ^[(Y )℄E = T(E)=Ker�(Y ) is alsoa generalization of the exterior algebra of E. For (Y ) = (Y N ), ^[(Y N )℄E willbe simply denoted by ^[N ℄E. One learly has ^[2℄E = ^2E = ^E for N = 2.39

In the ase N = 3, it an be shown that Ker(�3) is the two-sided ideal ofT(E) generated by theX Y Z + Z X Y + Y Z Xand the X Y X Xfor X;Y;Z 2 E. This implies that one has�3(X)�3(Y )�3(Z) + �3(Z)�3(X)�3(Y ) + �3(Y )�3(Z)�3(X) = 0and �3(X)�3(Y )(�3(X))2 = 0for any X;Y;Z 2 E and that these are the only independent relations inthe asso iative algebra Im(�3) = ^[3℄E. This means that ^[3℄E is the asso- iative unital algebra generated by the subspa e E with relations XY Z +ZXY + Y ZX = 0 and XYX2 = 0 for X;Y;Z 2 E. The graduation isindu ed by giving the degree one to the elements of E whi h is onsistentsin e the relations are homogeneous. It is lear on this example that thehomogeneous subspa es ^p[N ℄E of ^[N ℄E are generally not irredu ible for the(obvious) a tion of GL(E). It is not hard to see that one has!0 ^[N ℄ E = ^NEwhere !0 is a generator (' 1l) of ^0NE ' R, that is !0 is a y li ve tor forthe a tion of ^[N ℄E on ^NE.Corresponding to the generalization ^[(Y )℄E of the exterior algebra thereis a generalization [(Y )℄(M) of di�erential forms on a smooth manifold Mwhi h is de�ned in a similar way as (Y )(M) was de�ned in Se tion 3. This40

[(Y )℄(M) is then a graded asso iative algebra and (Y )(M) is a right graded[(Y )℄(M)-module (et .). In the ase (Y ) = (Y N ) one writes [N ℄(M) for thisgeneralization. For M = RD one has[N ℄(RD) = ^[N ℄RD� C1(RD)and, by identifying [N ℄(RD) as a graded-subspa e of T(RD�)C1(RD) andby using the anoni al at torsion-free linear onne tion of RD one an de�nea N -di�erential d on [N ℄(RD) by appropriate proje tion. One an pro eedsimilarity for [(Y )℄(RD) when (Y ) satis�es the assumption of Lemma 1 (orLemma 2, Lemma 7). More pre isely, the N - omplexes onstru ted so farare parti ular ases of the following general onstru tion.Let A = �n2NAn be an asso iative unital graded algebra generated by Delements of degree one �� for � 2 f1; : : : ;Dg su h thatXp2SN ��p(1) : : : ��p(N) = 0 (39)for any �1; : : : ; �N 2 f1; : : : ;Dg. Then the algebraA(RD) de�ned byA(RD) =AC1(RD) is a graded algebra and one de�nes a N -di�erential d on A(RD),i.e. a linear mapping d of degree one satisfying dN = 0, by settingd(a f) = (�1)na�� ��f (40)for a 2 An and f 2 C1(RD). Let M = �nMn be a graded right A-module,then M(RD) = M C1(RD) is a graded spa e whi h is a graded rightA(RD)-module and one de�nes a N -di�erential d on M(RD) by settingd(m f) = (�1)nm�� ��f (41)for m 2 Mn and f 2 C1(RD). The (irrelevant) sign (�1)n in formulas (40)and (41) is here in order to re over the usual exterior di�erential in the ase41

where A = ^RD� =M.It is lear that [N ℄(RD) = A(RD) for A = ^[N ℄RD� and that N (RD) =M(RD) for M = ^NRD�. If (Y ) satis�es the assumption of Lemma 1 one an take (in view of Lemma 7) A = ^[(Y )℄RD� and M = ^(Y )RD� and then[(Y )℄(RD) = A(RD) and (Y )(RD) =M(RD).8 Further remarksOur original unpublished proje t for proving Theorem 1 was based on the onstru tion of generalized algebrai homotopy in appropriate degrees. Letus explain what it means, why it is rather umbersome and why the proofgiven here, based on the introdu tion of the multigraded di�erential algebraA, is mu h instru tive and general and is related to the ansatz of Green forthe fermioni parastatisti s of order N � 1 (in the ase dN = 0).Let = �nn be a N - omplex (of o hains) with N -di�erential d. Analgebrai homotopy for the degree n will be a family of N linear mappingshk : n+k ! n+k�N+1for k = 0; : : : ; N � 1 su h that PN�1k=0 dN�1�khkdk is the identity mapping Inof n onto itself. If su h a homotopy exists for the degree n, then one hasHn(k) = 0 for k 2 f0; : : : ; N � 1g. Indeed let ! 2 n be su h that dk! = 0then one has ! = dN�k �Pk�1p=0 dk�1�phpdp!�.Our original strategy for proving Theorem 1 was to show that one an onstru t indu tively su h homotopies for the degrees (N � 1)p with p � 1in the ase of the N - omplex N (RD) and our idea was to exhibit expli it42

formulas. Unfortunately this latter task seems very diÆ ult in general. Weonly su eeded in produ ing formulas in a losed form in the ase N = 3and we refrain to give them here be ause this would imply explanations ofour normalization onventions whi h have no hara ter of naturality. ThediÆ ulty is indeed a problem of normalization. For the lassi al ase N = 2,one obtains a homotopy formula by using the inner derivation ix with respe tto the ve tor �eld x with omponents x�. In this ase one uses the fa t thatboth d and ix are antiderivations and that the Lie derivative Lx = dix+ixd isthe sum of the form-degree and of the degree of homogeneity in x. This giveshomotopy formulas for forms whi h are homogeneous polynomials in x andone gets rid of the above degree by appropriately weighted radial integrationand obtains thereby the usual homotopy formula for positive form-degree. Inthis ase the normalizations are �xed by the (anti)derivation properties. Inthe ase N � 3, d has no derivation property and one has to generalize ixwhi h is possible with iNx = 0 but there is no natural normalization sin e ix annot possess derivation property. As a onsequen e the appropriate gen-eralization of the Lie derivative involves a linear ombination of produ ts ofpowers of d and ix with oeÆ ients whi h have to be �xed at ea h step. Thatthis is possible onstitutes a umbersome proof of Theorem 1 but does notallow easily to write losed formulas.The interest of the proof of Theorem 1 presented here lies in the fa tthat it follows from the more general Theorem 2 whi h an be applied toother situations in parti ular to investigate the generalized ohomology of[N ℄(RD). Moreover, the realization of N (RD) embedded in A is related tothe Green ansatz for the parafermioni reation operators of order N � 1.Indeed if instead of equipping A with the graded ommutative produ t one43

repla es in the de�nition of A the graded tensor produ ts of graded alge-bras by the ordinary tensor produ ts of algebras (applying the appropriateKlein transformation) then the dix� and the djx� ommute for i 6= j andthe di de�ned by the same formulas (14) ommute, i.e. satisfy didj = djdiinstead of (15), from whi h it follows that Pi di is only a N -di�erential.This latter N -di�erential indu es the N -di�erential of N (RD) � A and therelation with the Green ansatz be omes obvious after Fourier transformation.The basi N - omplexes onsidered in this paper are N - omplexes ofsmooth tensor �elds on RD and we have seen the diÆ ulty to extend the for-malism on an arbitrary D-dimensional manifoldM . In the ase of a omplex(holomorphi ) manifoldM of omplex dimension D, there is an extension ofthe previous formalism at the ��-level whi h we now des ribe shortly.Let M be a omplex manifold of omplex dimension D and let T be asmooth ovariant tensor �eld of type (0; p) (i.e. of d�z-degree p) with lo- al omponents T��1:::��p in lo al holomorphi oordinates z1; : : : ; zD. Then���p+1T��1:::��p are the omponents of a well-de�ned smooth ovariant tensor�eld �rT of type (0; p + 1) sin e the transition fun tions are holomorphi ,where ��� denotes the partial derivative �=��z� of smooth fun tions. Let(Y ) be a sequen e (Yp)p2N of Young diagrams su h that jYpj = p (8p 2 N)and denote by 0;p(Y )(M) the spa e of smooth ovariant tensor �elds of type(0; p) with Young symmetry type Yp (with obvious notation). Let us set0;�(Y )(M) = �p0;p(Y )(M) and generalize the ��-operator by setting�� = (�1)pYp+1 Æ �r : 0;p(Y )(M)! 0;p+1(Y ) (M)with obvious onventions. It is lear that if (Y ) is su h that for any p 2 Nthe number of olumns of Yp is stri tly less than N , then one has ��N = 044

so 0;�(Y )(M) is a N - omplex (for ��). In parti ular one has the N - omplex0;�N (M) for �� by taking (Y ) = (Y N ). One has an obvious extension ofTheorem 1 ensuring that the generalized ��- ohomology of 0;�N (C D ) vanishesin degree (N � 1)p (i.e. bidegree or type (0; (N � 1)p)) for p � 1. It isthus natural to seek for an interpretation of this generalized ohomology for0;�N (M) in degrees (N � 1)p with p � 1 for an arbitrary omplex manifoldM and one may wonder whether it an be omputed in terms of the ordinary��- ohomology of M .Referen es[1℄ C. Be hi, A. Rouet, R. Stora. Renormalization models with brokensymmetries. in \Renormalization Theory", Eri e 1975, G. Velo, A.S.Wightman Eds, Reidel 1976.[2℄ H. Boerner. Representations of groups. North Holland 1970.[3℄ N. Boulanger, T. Damour, L. Gualtieri, M. Henneaux. In onsisten y ofintera ting, multigraviton theories. Nu l. Phys. B597 (2001) 127-171.[4℄ B. de Wit, D.Z. Freedman. Systemati s of higher-spin gauge �elds. Phys.Rev. D21 (1980) 358-367.[5℄ T. Damour, S. Deser. Geometry of spin 3 gauge theories. Ann. Inst. H.Poin ar�e 47 (1987) 277-307.[6℄ M. Dubois-Violette. Generalized di�erential spa es with dN = 0 and theq-di�erential al ulus. Cze h J. Phys. 46 (1997) 1227-1233.[7℄ M. Dubois-Violette. dN = 0 : Generalized homology. K-Theory 14(1998) 371-404. 45

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