A combinatorial method for computing Steenrod squares

A COMBINATORIAL METHOD FOR COMPUTING STEENROD SQUARES Roc´ıo Gonz´alez-D´ıaz and Pedro Real Dpto. de Matem´atica Aplicada I Facultad de Inform´atica y Estad´ıstica Universidad de Sevilla rogo[email protected], [email protected] Abstract We present here a combinatorial method for computing cup-iproducts and Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X. A generalization of this method to Steenrod reduced powers is sketched. Both authors are partially supported by the PAICYT research project FQM-0143 from Junta de Andaluc´ıa and the DGESIC research project PB97-1025-C02-02 from Education and Science Ministry (Spain). 1 1 Introduction Cohomology operations are algebraic operations on the cohomology groups of spaces commuting with the homomorphisms induced by continuous mappings. This machinery is useful when the graded vector space structure and the cup product on cohomology fail to distinguish two spaces by their cohomology. Steenrod squares [20] constitute an extremely important class of cohomology operations not only in Algebraic Topology but also in the area of simplicial methods in Homological Algebra (cohomology of groups, Hochschild cohomology of algebras, ...). We here study in detail the underlying combinatorial structures of the definition of these cohomology operations. More concretely, we will move in the framework of Simplicial Topology [12] in which the basic objects are simplicial sets, that is, graded sets endowed with face (∂i) and degeneracy (si) operators, satisfying several commutativity relations. Roughly speaking, a simplicial set can be considered as an algebraic generalization of the structure of a triangulated polyhedron although the former features a more rigid combinatorial structure than the latter. In this context, we develop a suitable setting in which all Steenrod cohomology operations can be studied simultaneously. It is well-known that there are several methods for constructing Steenrod squares. One of them consists of making these operations using the cohomology of Eilenberg-Mac Lane spaces (see, for instance, [[12]; p. 107]). Another method is determined by the construction of a family of morphisms {Di}measuring the lack of commutativity of the cup product on the cochain level [20]. This sequence of morphisms is called higher diagonal approximation. Its existence is always guaranteed by the acyclic models method [5]. In this way, it is possible to derive a recursive procedure to obtain the explicit formula for any Di(see [4], [[13]; Sect. 7]). In this paper, we present an alternative method for obtaining the explicit formula for a higher diagonal approximation. In [15], the formula for Diis established in terms of the component morphisms of a given Eilenberg-Zilber contraction (a special homotopy equivalence) from CN ∗(X×X) onto CN ∗(X)⊗CN ∗(X), where CN ∗(X) denotes the normalized chain complex of a given simplicial set X. Now, we have the following problems. On one hand, the component morphisms of the contraction above are defined in terms of face and degeneracy operators of the simplicial set X. On the other hand, the formula for a morphism Dialways involves the use of the homotopy operator of the Eilenberg-Zilber contraction and the explicit formula for this last morphism is determined by shuffles (a special type of permutation) of degeneracy operators. In consequence, if we try to express in 2 this way the morphisms Diin terms of face and degeneracy operators of X, the number of summands appearing in the formula for a morphism Dievaluated over an element of degree nis, in general, at least 2n. Therefore, an algorithm that would be designed starting from these formulae would be too slow for practical implementation. Because of this, the idea of simplifying these formulae arises in a natural way. This simplification or normalization is based on the fact that any composition of face and degeneracy operators of the simplicial set Xcan be put in a “canonical” way. That is, a composition of this type can be expressed in the unique form: sjt· · · sj1∂i1· · · ∂is, where jt>· · · > j1≥0 and is>· · · > i1≥0. Moreover, taking into account that the image of a morphism Dilies in CN ∗(X)⊗CN ∗(X), those summands of the simplified formula for Diwith a factor having a degeneracy operator in its expression, can be eliminated. The reason is that this factor applied to an element of the simplicial set Xis zero in the normalized chain complex associated to X. In this way, we here obtain more simple formula for the morphism Di. In this way, the classical definition of Steenrod squares: Sqi(c)(x) = (µ(< c ⊗c, Dj−i(x)>), i ≤j 0, i > j (1) where c∈Hom(CN j(X),Z2), x∈CN i+j(X) and µis the homomorphism induced by the multiplication in Z2, is complemented by a manageable combinatorial formulation of the higher diagonal approximation. As we will remark later, this description can be considered as a direct translation of the most ancient definition of Steenrod squares (see [18]) to the general setting of the Simplicial Topology. We give a more detailed explanation in the third section. Still, we think that this combinatorial machinery could be substantially improved in the future by exploiting the well-known properties of Steenrod squares [1, 20] and advanced techniques for calculating cocycles (see, for example, [9, 10]). In this way, a “reasonably efficient” algorithm computing the cohomology algebra of several important simplicial sets could be derived. Finally, we start an analogous study to that given in [15] for Steenrod reduced powers. More precisely, we provide a simplicial description of these operations in terms of the component morphisms of an Eilenberg-Zilber contraction from 3 CN ∗(X×ptimes · · · ×X) to CN ?(X)⊗ptimes · · · ⊗CN ∗(X), and a 0-sequence {γi}i≥0in the symmetric group Gp. Here is a summary of the present paper. Section 2 is dedicated to notation, terminology and a presentation of the problem. In Section 3, we show an explicit combinatorial definition of cup-iproducts and, consequently, of Steenrod squares. In Section 4, we study Steenrod reduced powers from this point of view. Finally, a proof of the main theorem enunciated in Section 3 is given in Section 5. We are grateful to Prof. Tom´as Recio for his helpful suggestions for making the exposition more readable and, hopefully, more informative. We also wish to acknowledge our debt to the referees for their many valuable indications. 2 The higher homotopy commutativity of the Alexander-Whitney operator The aim of this section is to give some preliminaries and a brief account of the work done in [15] in order to facilitate the understanding of the rest of the paper. Most of the material given in this section can be found in [12], [11], [17] and [21]. Asimplicial set Xis a sequence of sets X0, X1, . . ., together with face operators ∂i:Xn→Xn−1and degeneracy operators si:Xn→Xn+1 (i= 0,1, . . . , n), which satisfy the following “simplicial” identities: (s1) ∂i∂j=∂j−1∂i,if i < j; (s2) sisj=sj+1si,if i≤j; (s3) ∂isj=sj−1∂i,if i < j, (s4) ∂isj=sj∂i−1,if i > j + 1, (s5) ∂jsj= 1X=∂j+1sj. The elements of Xnare called n-simplices. A simplex xis degenerated if x=siy for some simplex yand degeneracy operator si; otherwise, xis non degenerated. The following elementary lemma will be essential in the proof of the main theorem of this paper. 4 Lemma 2.1 [12] Any composition µ:Xm→Xnof face and degeneracy operators of a simplicial set Xcan be put in a unique “canonical” form: sjt· · · sj1∂i1· · · ∂is, where n>jt>· · · > j1≥0,m≥is>· · · > i1≥0and n−t+s=m. Let Rbe a ring which is commutative with unit. A chain (resp. cochain)complex M={Mn, dn}(resp. C={Cn, δn}) is a graded (over the integers) R-module together with a R-module map dof degree −1 (resp. of degree +1), called the differential, such that dndn+1 = 0 (resp. δn+1 δn= 0). An element cof Cnis called n-cochain. The homology H∗(M) (resp. the cohomology H∗(C) is the family of modules Hn(M) = Ker dn/Im dn+1 (resp. Hn(C) = Ker δn/Im δn−1). If Mis a chain complex over Rand Gis an R-module, there is a cochain complex Hom(M, G) = {Hom(Mn, G), δn}, where, if c∈Hom(Mn, G), then δnc∈Hom(Mn+1, G) is defined by (δnc)(x) = c(dn+1(x)), x ∈Mn+1. We also write < c, x > instead of c(x) and set < c, x >= 0 if the degree of the cochain cis not equal to the degree of the element x. In this notation, < δnc, x > =< c, dn+1(x)> . Now, given a simplicial set X, let C∗(X) denotes the chain complex {Cn(X), dn}, in which Cn(X) is the free R-module generated by Xnand dn:Cn(X)→Cn−1(X) is defined by dn= n X i=0 (−1)i∂i. Let us denote by s(C∗(X)) the graded R-module generated by all the degenerated simplices. In C∗(X), we have that dn(s(Cn−1(X))) ⊂s(Cn−2(X)) and, then CN ∗(X) = {Cn(X)/s(Cn−1(X)), dn}is a chain complex called the normalized chain complex associated to X. Given a R-module G, let us define by C∗(X;G) the cochain complex associated to CN ∗(X). In this way, we define the homology and cohomology of Xwith coefficients in a 5 R-module Gby H∗(X;G) = H∗(CN ∗(X)⊗G) and H∗(X;G) = H∗(C∗(X;G)), respectively. Eilenberg and Mac Lane defined in [6] a contraction of chain complexes from N onto M, as a triple (f, g, φ) in which f:N→M(projection) and g:M→N (inclusion) are chain maps, and φ:N→N(homotopy operator) is a map of R-module raising degree by 1. Moreover, it is required that (c1) fg = 1M,(c2) φd +dφ +gf = 1N, (c3) φg = 0,(c4) fφ = 0,(c5) φφ = 0. Hence, this definition implies that the “big” complex Nis homology equivalent to the “small” complex Min a strong way. In fact, a contraction is a special homotopy equivalence between chain complexes. If we have two contractions (fi, gi, φi) from Nito Mi, with i= 1,2 then, the following contractions can be constructed (see [6]): •The tensor product contraction (f1⊗f2, g1⊗g2, φ1⊗g2f2+ 1N1⊗φ2) from N1⊗N2to M1⊗M2. •If N2=M1, the composition contraction (f2f1, g1g2, φ1+g1φ2f1) from N1to M2. If pand qare non-negative integers, a (p, q)-shuffle (α, β) is a partition of the set {0,1, . . . , p +q−1}of integers into two disjoint subsets, α1<· · · < αpand β1<· · · < βq, of pand qintegers, respectively. The signature of the shuffle (α, β) is defined by sig(α, β) = p X i=1 αi−(i−1). After these preliminaries, we are able to describe a very important homotopy equivalence in Algebraic Topology. This contraction tells us that the associated chain complex CN ∗(X×Y) reduces to the tensor product of chain complexes CN ∗(X) and CN ∗(Y). An Eilenberg-Zilber contraction [8] from CN ∗(X×Y) to CN ∗(X)⊗CN ∗(Y), where Xand Yare given simplicial sets, is defined by the triple (AW, EML, SHI) where: 6 •The Alexander-Whitney operator AW :CN ∗(X×Y)−→ CN ∗(X)⊗CN ∗(Y) is defined by: AW(am×bm) = m X i=0 ∂i+1 · · · ∂mam⊗∂0· · · ∂i−1bm. If X=Y,AW can be considered as a “simplicial approximation” to the diagonal and this operator provides a method for constructing the cup product in cohomology. If we interchange the factors amand bmin the formula, we obtain a different approximation. Comparison of these two different approximations to the diagonal leads to the Steenrod squares. •The Eilenberg-Mac Lane operator EML :CN ∗(X)⊗CN ∗(Y)−→ CN ∗(X×Y) is defined by: EML(ap⊗bq) = X (α,β)∈{(p,q)−shuffles} (−1)sig(α,β)sβq· · · sβ1ap×sαp· · · sα1bq. This operator can be seen as a process of “triangulation” in the cartesian product X×Y. •And the Shih operator SHI :CN ∗(X×Y)−→ CN ∗+1(X×Y) is defined by: SHI(a0×b0) = 0; SHI(am×bm) = X(−1)¯m+sig(α,β)+1 sβq+ ¯m· · · sβ1+ ¯ms¯m−1∂m−q+1 · · · ∂mam ×sαp+1+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1bm; where ¯m=m−p−q,sig(α, β) = p+1 X i=1 αi−(i−1), and the last sum is taken over all the indices 0 ≤q≤m−1, 0 ≤p≤m−q−1 and (α, β)∈ {(p+ 1, q)-shuffles}. A recursive formula for the SHI operator has already been given by Eilenberg and Mac Lane in [7]. The explicit formula given here was stated by Rubio in [16] and proved by Morace in the appendix of [14]. In contrast to the deep studies found in the literature on the AW and EML operators, it turns to be quite surprising the lack of interest shown up to now in the study of the homotopy operator involved in an Eilenberg-Zilber contraction, not only from the point of view of getting its explicit formula but also of obtaining algebraic preservation results of this operator with regard to the underlying coalgebra structure on CN ∗(X×X). 7 Given a simplicial set Xand a positive integer p, we can form a contraction (f, g, φ) from CN ∗(X×ptimes · · · ×X) to CN ∗(X)⊗ptimes · · · ⊗CN ∗(X), appropriately composing Eilenberg-Zilber contractions. If p= 2, then (f, g, φ) is the contraction (AW, EML, SHI). The contraction from CN ∗(X×X×X) onto CN ∗(X)⊗CN ∗(X)⊗CN ∗(X) is defined by the composition of the Eilenberg-Zilber contraction from CN ∗(X×X×X) to CN ∗(X)⊗CN ∗(X×X) and the tensor product contraction of the identity morphism 1CN ∗(X)and the Eilenberg-Zilber contraction from CN ∗(X×X) onto CN ∗(X)⊗CN ∗(X). And so on. From now on, the contractions obtained in this way will also be called Eilenberg- Zilber contractions. Let pbe a positive integer. Let us define several chain maps (or morphisms) we use in this paper. We will omit in the notation of these morphism its dependency on p. The diagonal map ∆ : CN ∗(X)→CN ∗(X×ptimes · · · ×X) is defined by ∆(x) = (x, ptimes . . . , x). The following automorphisms are also defined: t:CN ∗(X×ptimes · · · ×X)→CN ∗(X×ptimes · · · ×X), such that t(x1, x2, . . . , xp) = (x2, . . . , xp, x1) and T:CN ∗(X)⊗ptimes · · · ⊗CN ∗(X)→CN ∗(X)⊗ptimes · · · ⊗CN ∗(X) defined by T(x1⊗x2⊗ · · · ⊗ xp) = (−1)|x1|(|x2|+···+|xp|)x2⊗ · · · ⊗ xp⊗x1. We now outline the problem concerning Steenrod squares in which we are interested. It is well-known that the AW operator is not commutative, that is, assuming that X=Y,AWt 6=AW. On the other hand, this operator determines the cup product in cohomology. If Gis a ring, given two cochains c∈Ci(X;G) and c0∈Cj(X;G), and x∈CN i+j(X), the cup-product of cand c0is defined by: c^c0(x) = µ(< c ⊗c0, AW ∆(x)>) =µ(< c, ∂i+1 · · · ∂i+jx > ⊗< c0, ∂0· · · ∂i−1x >), where µis the homomorphism induced by the multiplication on G. Steenrod in [19] determined that there exists an infinite sequence of morphisms {Di}, called higher diagonal approximation, which “measures” this lack of commutativity. More precisely, there is a sequence of graded homomorphisms Di:CN ∗(X)→CN ∗(X)⊗CN ∗(X) of degree isuch that: 8 D0=AW ∆ d⊗Di+1 + (−1)iDi+1 d=T Di+ (−1)i+1Di, where dand d⊗are the differentials of CN ∗(X) and CN ∗(X)⊗CN ∗(X), respectively. Moreover, the morphism Dican be expressed in the form Di=hi∆, where hi:CN ∗(X×X)→CN ∗(X)⊗CN ∗(X) is a homomorphism of degree i. In the literature, the existence of the tower of iterated Alexander-Whitney operators {hi}is guaranteed by the acyclic models method (see [5]). This technique can be considered as a constructive method in the simplicial category and recursive formulae for {Di}can be established (see [13]). In [15], a different approach is presented. The strong homotopy commutativity of AW is determined by making use of the explicit Eilenberg-Zilber contraction (AW, EML, SHI). Hence, the formula for a morphism Diis given in terms of the morphisms AW,SHI, the diagonal ∆ and the automorphism t. More precisely, the formula for hiis AW (t SHI)i, for all i∈N. Now, the definition of the cohomology operation Sqi:Hj(X;Z2)→ Hj+i(X;Z2) (see (1)) takes the form: Sqi(c)(x) = (µ(< c ⊗c, AW (t SHI)j−i(x, x)>), i ≤j 0, i > j (2) where c∈Cj(X;Z2) and x∈CN i+j(X). It is obvious that the formulae for the morphism hican be given in terms of face and degeneracy operators of X. Our objective in the next section is to show how to “simplify” these formulae and to obtain an explicit definition of Steenrod squares only in terms of face operators of X. 3 An explicit combinatorial description of the cup-iproducts It is clear that the image of hilies in CN ∗(X)⊗CN ∗(X). Therefore, if we express the factors of the summands of the formula for Diin a canonical way (see Lemma 2.1), those summands of the simplified formula for Dihaving a factor with a degeneracy operator in its expression must be eliminated. 9 Proposition 4.1 Let pbe prime with p≥2and ia non-negative integer, then φdγi· · · φγ1φ= (−1)i−1φγi· · · φγ1dφ + i−1 X k=1 (−1)i−kφγi· · · φγk+2φγφγk−1· · · φγ1φ. Proof We prove the proposition by induction on the parameter i. •If i= 1 then, φdγ1φ=φγ1dφ. •If i= 2 then, φdγ2φγ1φ=φγ2dφγ1φ=φγ2(gf −1−φd)γ1φ =−φγφ −φγ2φγ1dφ. •In general, φdγiφγi−1· · · φγ1φ=φγidφγi−1· · · φγ1φ =φγi(gf −1−φd)γi−1· · · φγ1φ =−φγφγi−2· · · φγ1φ−φγiφdγi−1· · · φγ1φ (by induction assumption) =−φγφγi−2· · · φγ1φ−(−1)i−2φγiφγi−1· · · φγ1dφ − i−2 X k=1 (−1)i−1−kφγiφγi−1· · · φγk+2φγφγk−1· · · φγ1φ = (−1)i−1φγi· · · φγ1dφ + i−1 X k=1 (−1)i−kφγi· · · φγk+2φγφγk−1· · · φγ1φ. 2 Let Γi(k) = fγiφγi−1· · · φγk+2φγφγk−1φ· · · φγ1φ∆. We can prove Proposition 4.2 Let pbe prime and ia non-negative integer, then fγiφγi−1· · · φγ2φ∆ = fγi−1φγi−2· · · φγ2φγ1φ∆ + i−1 X k=1 (−1)k+1Γi(k). 16 Proof Using γk+1 =γk+ (−1)k+1γ, for all 1 < k < i, we have fγiφγi−1· · · φγk+2φγk+1φγk−1· · · φγ1φ∆ =fγiφγi−1· · · φγk+2φγkφγk−1· · · φγ1φ∆+(−1)k+1Γi(k). If we use this fact successively, we obtain the following identity: fγiφγi−1· · · φγ2φ∆ = fγi−1φγi−2· · · φγ2φγ1φ∆ + i−1 X k=1 (−1)k+1Γi(k) 2 The main result of this section is the following one. Theorem 4.3 Let pbe a prime number, p≥2, and ia non-negative integer. Then, there exists a sequence of morphisms {Di}defined by Di=fγiφγi−1· · · φγ1φ∆, verifying that dDi+ (−1)i+1Did=αiDi−1, where α2j−1=T−1and α2j= 1 + T+T2+· · · +Tp−1. Proof Let us begin with the first term of the identity: dDi+ (−1)i+1Did=dfγiφγi−1· · · φγ1φ∆+(−1)i+1fγiφγi−1· · · φγ1φ∆d =fγidφγi−1· · · φγ1φ∆ +(−1)i+1fγiφγi−1· · · φγ1(gf −1−dφ)∆ =fγidφγi−1· · · φγ1φ∆ +(−1)ifγiφγi−1· · · φγ2φ∆+(−1)ifγiφγi−1· · · φγ1dφ∆ (by Proposition 4.1, we get) =fγidφγi−1· · · φγ1φ∆+(−1)ifγiφγi−1· · · φγ2φ∆ +(−1)i(−1)i−2fγiφdγi−1· · · φγ1φ∆ + i−2 X k=1 (−1)i(−1)kfγiφγi−1· · · φγk+2φγφγk−1φ· · · φγ1φ∆ 17 =fγi(dφ +φd)γi−1· · · φγ1φ∆ +(−1)ifγiφγi−1· · · φγ2φ∆ + i−2 X k=1 (−1)i−kΓi(k) =fγi(gf −1)γi−1· · · φγ1φ∆ +(−1)ifγiφγi−1· · · φγ2φ∆ + i−2 X k=1 (−1)i−kΓi(k) (let β2j=T+T2+· · · +Tp−1and β2j−1=T, then we have) =βifγi−1φγi−2· · · φγ1φ∆ +(−1)ifγiφγi−1· · · φγ2φ∆ + i−1 X k=1 (−1)i−kΓi(k) (Proposition 4.2 implies) =βifγi−1φγi−2· · · φγ1φ∆+(−1)ifγi−1φγi−2· · · φγ1φ∆ + i−1 X k=1 (−1)i+k+1Γi(k) + i−1 X k=1 (−1)i−kΓi(k) =αifγi−1φγi−2· · · φγ1φ∆. 2 5 Proof of the main theorem The proof consists in finding out the factors of the formula (written in the canonical way) that are degenerated and in eliminating the summands having these factors. First of all, notice that using the commutativity properties of the operators of a simplicial set (essentially, (s3)), it is easy to see that a factor of the formula whose expression begins (on the left) by ∂j1· · · ∂jtsk· · · (6) such that 0 ≤j1<· · · < jt< k, is degenerated in its simplified form. Having said that, let us begin with the proof of the theorem. For n= 0, we obtain the explicit formula for the Alexander-Whitney operator. Let us assume that the formula is true for k≤nso, let us prove that the formula is true for the case n+ 1. 18 Let us consider that nis even (when nis odd, the proof is similar). In this case, by induction assumption, the formula over an element of degree mis as follows: AW (t SHI)n+1 =AW (t SHI)n(t SHI) = m+1 X in=n · · · i1−1 X i0=0 X(−1)A(n)+B(n,m+1,¯ı)+C(n,m+1,¯ı)+D(n,¯ı)(−1)¯m+sig(α,β)+1 ∂i0+1 · · · · ∂m+1sαp+1+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1 ⊗∂0· · · · ∂in−1sβq+ ¯m· · · sβ1+ ¯ms¯m−1∂m−q+1 · · · ∂m where ¯ı = (i0, i1, . . . , in), ¯m=m−p−q,sig(α, β) = p+1 X i=1 αi−(i−1), and the last sum is taken over all the indices 0 ≤q≤m−1, 0 ≤p≤m−q−1 and (α, β)∈ {(p+ 1, q)-shuffles}. Let us recall that if the formula is in the normalized form, the summands which have a degenerated factor must be eliminated. The following cases are considered: If in> m −p, we have to consider the following cases: •If in=m−p+tand βq< q −1 + t, with 1 ≤t≤pthen, αp+1 =p+q > · · · > αt+1 =q+t > αt=q+t−1> βq. So, the first factor of these summands is: ∂i0+1 · · · · ∂in−1−1∂m−p+t+1 · · · ∂m+1sm· · · sm−p+t−1sαt−1· · · sα1∂¯m· · · ∂m−q−1 =∂i0+1 · · · · ∂in−1−1sm−p+t−1sαt−1· · · sα1∂¯m· · · ∂m−q−1. Since in−1−1< in−1 = m−p+t−1, this factor is degenerated by (6). •If in=m−p+tand βq=q−1 + t, with 1 ≤t≤pthen, αp+1 =p+q > · · · > αt+1 =q+t > βq=q+t−1. And hence, in this case, the expression of the first factor begins in the form: ∂i0+1 · · · · ∂in−1−1∂m−p+t+1 · · · ∂m+1sm· · · sm−p+tsαt+ ¯m· · · =∂i0+1 · · · · ∂in−1−1sαt+ ¯m· · · . 19 Now, we have to consider two different cases: –If in−1−1< αt+ ¯mthen, this factor is degenerated. –If in−1−1≥αt+ ¯m, let us denote αt=q+t−1−j, where 1 ≤j≤q+t−1. Then, αp+1 =p+q > · · · > αt+1 =q+t, and βq=q+t−1>· · · > βq−j+1 =q+t−j > αt=q+t−1−j. Hence, the second factor of these summands is: ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p+t−1sm−p+t−1· · · sm−p+t−jsβq−j+ ¯m· · · =∂0· · · · ∂in−2−1sin−1· · · sm−p+t−jsβq−j+ ¯m· · · Since in−2−1< in−1, this factor is degenerated. •If in=m−p+tand βq> q−1+t, with 1 ≤t≤pthen in−1< βq+ ¯m. So, the second factor of the summands has the form (6) and hence, these summands must be eliminated. •If in=m+1 and βq< p+qthen αp+1 =p+qand since in−1−1< in−1 = m then, the first factor of the summands is degenerated. •If in=m+ 1 and βq=p+qthen βq=p+q > · · · > βj+1 =p+j+ 1 > αp+1 =p+j with 0 ≤j≤q−1 and the first factor of the summands is: ∂i0+1 · · · · ∂in−1−1sm−q+jsαp+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1. We have to consider two different cases: –If in−1−1< m −q+jthen, this factor is degenerated. –If in−1−1≥m−q+jthen the second factor of the summands is: ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂msm· · · sm−q+j+1sβj+ ¯m· · · =∂0· · · · ∂in−2−1sin−1· · · sm−q+j+1sβj+ ¯m· · · , which is degenerated due to the fact that in−2−1< in−1. 20 If in< m −p, then in−1< βq+ ¯m. So, these summands have the second factor in the form (6) and hence, must be eliminated. If in=m−p, two cases hold: •If βq> q −1, then the second factor of these summands is degenerated as above. •If βq=q−1 and in−1>¯m−2 then αp+1 =p+q > · · · > α1=q > βq=q−1>· · · > β1= 0, and the second factor of the tensor product is ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p−1sm−p−1· · · sm−p−q−1∂m−q+1 · · · ∂m =∂0· · · · ∂in−2−1sin−1· · · sm−p−q−1∂m−q+1 · · · ∂m; and since in−2−1< in−1, this factor is degenerated. Finally, if βq=q−1 and in−1≤¯m−2 then the formula (save for the signs) corresponding to AW (t SHI)n+1 is: ¯m−2 X in−1=n−1 · · · i1−1 X i0=0 m−1 X q=0 m−q−1 X p=0 ∂i0+1 · · · · ∂in−1−1∂m−p+1 · · · ∂m+1sm· · · sm−p∂¯m· · · ∂m−q−1 ⊗∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p−1sm−p−1· · · s¯m−1∂m−q+1 · · · ∂m = m X i0 n+1=n+1 i0 n+1−1 X i0 n=n i0 n−1 X in−1=n−1 · · · i1−1 X i0=0 ∂i0+1 · · · · ∂in−1−1∂i0 n+1 · · · ∂i0 n+1−1 ⊗∂0· · · · ∂in−2−1∂in−1+1 · · · ∂i0 n−1∂i0 n+1+1 · · · ∂m; where i0 n= ¯m−1 and i0 n+1 =m−q. Now, let us study the signs of the formulae in this last case. Keeping in mind that we are working with the exponent of (−1), all the identities are mod 2. This proof is based on the fact that if and only if in=m−pand βq=q−1, the summands of AW (t SHI)nare non-degenerated. 21 We first verify the formula in the case n= 1. The exponent of (−1) associated to each summand is: ¯m+sig(α, β) + 1 = ¯m+ p+1 X i=1 αi−(i−1) + 1 = ¯m+ p+1 X i=1 (q−1 + i−(i−1)) + 1 = ¯m+q(p+1)+1 (note i0= ¯m−1 and i1=m−q) =i0+ (m+i1)(i0+i1) = A(1) + B(1, m,¯ı) + C(1,¯ı) + D(1, m,¯ı) where ¯ı = (i0, i1), as asserted. In general, we have to prove that A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + D(n, m + 1,¯ı) + ¯m+sig(α, β)+1 =A(n+ 1) + B(n+ 1, m,¯ı) + C(n+ 1,¯ı) + D(n+ 1, m,¯ı). •If nis even then D(n, m+1,¯ı) = 0 and the exponent of (−1) in each summand is: A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + ¯m+sig(α, β)+1 =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) +(m+p+in−1)(in−1+· · · +i0) + ¯m+q(p+1)+1 (since i0 n= ¯m−1, i0 n+1 =m−qand these identities are mod 2 then) =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) +(m+1+i0 n+1 +i0 n+in−1)(in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) + in−1+· · · +i0 +(i0 n+in−1)(in−1+· · · +i0)+(m+i0 n+1)(in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 +C(n+ 1,¯ı) + D(n+ 1, m,¯ı). 22 We have to distinguish two cases: –If n≡0 mod 4 then A(n) = A(n+ 1) and A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+1 +i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+i0 n=A(n+ 1) + B(n+ 1, m,¯ı). –If n≡2 mod 4 then A(n) = A(n+ 1) + 1 and A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 =A(n+1)+1+ n−2 2 X j=0 i2j+m+p+i0 n+in−1+· · · +i0 =A(n+1)+1+ n−2 2 X j=0 i2j+m+i0 n+1 +i0 n+ 1 +i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+1 +i0 n+1 +m=A(n+ 1) + B(n+ 1, m,¯ı). •If nis odd then the exponent of (−1) is: A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + D(n, m + 1,¯ı) + ¯m+sig(α, β)+1 =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(m+1+m+p)(m+p+in−1+· · · +i0) + ¯m+q(p+1)+1 (since i0 n= ¯m−1, i0 n+1 =m−qand these identities are mod 2 then) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(i0 n+1 +i0 n)(m+1+i0 n+1 +i0 n+in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + (i0 n+1 +i0 n)(i0 n+in−1+· · · +i0) +i0 n+1 +i0 n+ (i0 n+1 +i0 n)(m+i0 n+1) + i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(i0 n+1 +i0 n)(i0 n+in−1+· · · +i0) + i0 n+1 =A(n) + B(n, m + 1,¯ı) + i0 n+1 +C(n+ 1,¯ı) + D(n+ 1, m,¯ı). 23 Since nis odd then n≡1,3,5,7 mod 8 and, in these cases, A(n) = A(n+ 1). 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