Permutation Resolutions for Specht Modules of Hecke Algebras

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1 Permutation Resolutions for Specht Modules of Hecke Algebras Robert Boltje Department of Mathematics University of California Santa Cruz, CA U.S.A. Filix Maisch Department of Mathematics Oregon State University Corvallis, OR U.S.A. October 17, 2011 (revised March 25, 2012) Abstract In [BH], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proven in the symmetric group case. Introduction In [BH], Hartmann and the first author constructed, for any composition λ of a positive integer r, a finite chain complex of modules for the group algebra RW of the symmetric group W on r letters over an arbitrary commutative ring R. The last module in this complex is the dual of the Specht module MR Subject Classification: 20C08. Keywords: Iwahori Hecke algebra, Specht module, permutation module, resolution 1

2 S λ and the other modules are permutation modules with point stabilizers given by Young subgroups of W. The construction of the chain complex was completely combinatorial and characteristic-free. It was conjectured that this chain complex is exact whenever λ is a partition. In other words the chain complex is conjectured to be a resolution of the (dual) of the Specht module by permutation modules. Partial exactness results were already established in [BH] and a full proof of the exactness was given recently by Yudin and Santana (see [SY]) by translating the construction via the Schur functor. Other permutation resolutions of Specht modules have been considered by Donkin in [Dn], Akin in [A], Zelevinskii in [Z], Akin and Buchsbaum in [AB1] and [AB2], Santana in [S], Woodcock in [W1] and [W2], Doty in [Dy] and Yudin in [Y]. See [BH, Section 6] for a more detailed comparison of these constructions with the construction in [BH]. The goal of this paper is to (a) lift the construction of the chain complex in [BH] for the group algebra RW to a chain complex for modules of the Iwahori Hecke algebra H R r,q for any integral domain R such that the specialization q = 1 reproduces the original chain complex; and (b) lift the partial exactness proofs from [BH] to the new construction. Both goals are achieved, with part (a) being relatively straightforward, but part (b) requiring much more subtle arguments. The paper is arranged as follows. In Section 1 we establish the necessary notation and recall results about the Hecke algebra H R r,q from [DJ]. In Section 2 we introduce a group of homomorphisms between permutation modules of H R r,q and show that such homomorphisms are closed under composition. This is used in Section 3 to construct the chain complex which generalizes the chain complex in [BH]. In Section 4 we prove that the chain complex in question is exact in degrees 1 and 0, for arbitrary partitions λ, and that it is exact everywhere, for all partitions of the form (λ 1, λ 2, λ 3 ) with λ 3 1, cf. Theorems 4.2, 4.4, 4.5. The very technical proof of a key lemma, Lemma 4.3, is postponed to Section 5. The authors are grateful to the referee for the careful reading and for pointing out a mistake in the proof of Proposition 2.2 in the first version of this paper. 2

3 1 Notation and Quoted Results Throughout this paper we denote by R an integral domain. Unadorned tensor products and homomorphism sets will be understood to be taken over R. Moreover, we fix a positive integer r and denote by W the symmetric group on the set {1,..., r}. Elements of W are composed like functions applied on the right in order to be consistent with [DJ]. Thus, W acts from the right on {1,..., r} and we write (i)w or iw for i {1,..., r} and w W. We set S := {(i, i + 1) i = 1..., r 1}, the standard choice of simple transpositions. The set of positive (resp. non-negative) integers will be denoted by N (resp. N 0 ). 1.1 The Hecke algebra H. For a unit q of R we denote by H = Hr,q R the Hecke algebra as introduced in [DJ]. It has an R-basis consisting of the elements T w, w W. The multiplication in H is uniquely determined by the following formulas for w W and s S: T ws, if l(ws) = l(w) + 1, i.e., if iw 1 < (i + 1)w 1, T w T s = qt ws + (q 1)T w, if l(ws) = l(w) 1, i.e., if iw 1 > (i + 1)w 1. The element T 1 is the identity element of H. By we denote the strong Bruhat order on W : for u, v W, one has u v if there exists a reduced expression u = s 1 s n of u and integers 1 i 1 < < i k n with v = s i1 s i2 s ik. While, for given v, w W, it is in general difficult to express T v T w in terms of the basis elements T x, x W, Lemma 2.1(ii) in [DJ] and a quick induction argument on l(w) imply that T v T w wu W RT vu (1) and that T v T w = T vw if l(vw) = l(v) + l(w). (2) 1.2 The H-modules M λ. We denote by Γ the set of compositions λ = (λ 1, λ 2,...) of r and by Λ the set of partitions of r. We define the dominance order on Γ by λ µ: e e λ i µ i for all e 1. i=1 i=1 3

4 This defines a partial order on Γ. Note that it differs from the dominance relation defined in [DJ] which is not a partial order and is defined by λ µ: µ λ, (3) where λ = (λ 1, λ 2,...) denotes the dual of λ, i.e., λ i := {k N λ k i}, for i N. Note that λ is always a partition. Let λ be a composition of r. One associates to λ the set partition P λ := {{1,..., λ 1 }, {λ 1 + 1,..., λ 1 + λ 2 },...} of {1,..., r} and the standard parabolic subgroup W λ consisting of all w W which fix P λ element-wise. Note that W λ is generated by the elements s = (i, i + 1) S with the property that i and i + 1 belong to the same element of P λ. More generally, a subgroup of W is called a parabolic subgroup if it arises as the stabilizer of an arbitrary set partition of {1,..., r}, i.e., if it is conjugate to a subgroup of the form W λ. Intersections of parabolic subgroups are again parabolic subgroups. The right H-submodule M λ of the regular module H is defined as M λ := x λ H with x λ := T w. w W λ For v W λ one has x λ T v = q l(v) x λ, (4) cf. [DJ, Lemma 3.2]. Every coset W λ w W λ \W has a unique element of smallest length. We denote the set of these distinguished coset representatives by D λ. For w W λ and d D λ one has l(wd) = l(w) + l(d) and T wd = T w T d. (5) The elements x λ T d, d D λ, form an R-basis of M λ, cf. [DJ, Lemma 3.2]. As usual one identifies compositions λ with Young diagrams and we say that λ is the shape of the corresponding Young diagram. A λ-tableau is a filling of the r boxes of the Young diagram of λ with the numbers 1, 2,..., r. We denote the set of λ-tableaux by T(λ) and usually denote elements of T(λ) by t. The λ-tableau which contains the entries 1, 2,..., r in ascending order is denoted by t λ. Thus, for λ = (1, 0, 2) one has t (1,0,2) =

5 This provides a standard numbering of the boxes of the Young diagram of λ which we will use later. The group W acts from the right on T(λ) by simply applying an element w W to the entries of the tableau t T(λ). This action is free and transitive, and it yields a bijection W T(λ), w t λ w. A λ-tableau t is called row-standard if its entries are increasing in each row from left to right. The row-standard λ-tableaux form a subset T rs (λ) of T(λ). Two λ-tableaux t 1 and t 2 are called row-equivalent if they arise from each other by rearranging elements within each row. We denote the rowequivalence class of t by {t} and the set of row-equivalence classes by T(λ). One has canonical bijections D λ T rs (λ) T(λ) (6) given by d t λ d and t {t}. Thus, the canonical basis x λ T d, d D λ, could also be parametrized by T rs (λ) or T(λ). For d D λ we denote by ε d Hom(M λ, R) the R-module homomorphism with the property that ε d (x λ T e ) = δ d,e for all e D λ. In other words, the elements ε d, d D λ, form the dual basis of the R-basis x λ T d, d D λ. If t T rs (λ) corresponds to d D λ under the canonical bijection in (6) we will also write ε t instead of ε d. For λ Λ we have an obvious bijection T(λ) T(λ ), t t, where t is the reflection of t with respect to the diagonal axis. Note that (tw) = t w for all t T(λ) and w W. 1.3 The homomorphisms ϕ λ,µ d : M µ M λ. Let µ, λ Γ be compositions of r. The elements of D λ,µ := D λ D 1 µ form a set of representatives of the double cosets W λ \W/W µ, and each element d D λ,µ is the unique element of shortest length in its double coset W λ dw µ. By [DJ, Theorem 3.4], the set D λ,µ parametrizes an R-basis ϕ λ,µ d, d D λ,µ, of Hom H (M µ, M λ ) given by ϕ λ,µ d (x µ) = T w = x λ T de = x λ T d T e, (7) w W λ dw µ e D ν W µ e D ν W µ where ν Γ is determined by W ν = d 1 W λ d W µ. The last equality follows from (5). We want to mention that the cited theorem requires that R is a 5

6 principal ideal domain. But a careful examination of the canonical isomorphisms in [DJ, Theorems 2.5, 2.6, 2.7] involved in the proof of the theorem shows that this hypothesis is unnecessary. (More precisely, the R-modules Hom H (M Hν, N T d ) in [DJ, Theorem 2.8] are free of rank one in the case that M and N are trivial modules.) We will use a combinatorial description of the set D λ,µ in terms of the set T(λ, µ) of generalized tableaux T of shape λ and content µ. Such a generalized tableau T is a filling of the r boxes of the Young diagram of shape λ with µ 1 entries equal to 1, µ 2 entries equal to 2, etc. We denote by Tµ λ T(λ, µ) the generalized tableau which has its boxes filled in the natural order. If we number the boxes of the Young diagram of λ according to the entries of t λ T(λ), we may view T(λ, µ) as the set of functions T : {1,..., r} N with the property T 1 (i) = µ i for all i N. The group W acts transitively on T(λ, µ) from the left by (wt )(i) := T (iw), for T T(λ, µ), w W and i {1,..., r}. The stabilizer of Tµ λ is equal to W µ. This defines a bijection W/W µ T(λ, µ), ww µ wtµ λ. We call two generalized tableaux T 1, T 2 T(λ, µ) row-equivalent if they arise from each other by rearranging the entries within the rows, i.e., if T 2 = wt 1 for some w W λ. If we denote the row equivalence class of T T(λ, µ) by {T } and the set of such classes by T(λ, µ) then we obtain a bijection W λ \W/W µ T(λ, µ), W λ ww µ {wtµ λ }. A generalized tableau T T(λ, µ) is called row-semistandard if in each of its rows the entries are in their natural order from left to right. We denote the set of these tableaux by T rs (λ, µ). Each row-equivalence class contains a unique row-semistandard element. Thus, T rs (λ, µ) T(λ, µ), T {T }, is a bijection. Altogether, we now have canonical bijections D λ,µ W λ \W/W µ T(λ, µ) T rs (λ, µ) (8) and we may use each of these sets to parametrize the basis ϕ λ,µ d, d D λ,µ, of Hom H (M µ, M λ ). So, if T T rs (λ, µ) and d D λ,µ correspond under the above bijection, we also write ϕ λ,µ T instead of ϕ λ,µ d. We leave it to the reader to check that d D λ,µ corresponds to T T rs (λ, µ) if and only if ϕ λ,µ T homomorphism defined in [BH, Subsection 1.5]. = θ T : M µ M λ in the case q = 1, with θ T the 1.4 The Specht modules S λ. Let λ Λ be a partition of r. One defines elements y λ := w W λ ( q) l(w) T w and z λ := x λ T wλ y λ M λ 6

7 of H, where w λ W is defined by the property that (t λ w λ ) = t λ. The Specht module S λ is defined by S λ := z λ H. Since z λ M λ, S λ is an H-submodule of M λ. A λ-tableau t T(λ) is called a standard λ-tableau if t T rs (λ) and t T rs (λ ). We denote the set of standard λ-tableaux by T st (λ). More generally, for µ Γ, we say that a generalized tableau T T(λ, µ) is standard if T T rs (λ, µ) and if the entries of T are strictly increasing along each column from top to bottom. We denote the set of standard λ-tableaux of content µ by T st (λ, µ). We extend the definition of S λ, T st (λ), and T st (λ, µ) to arbitrary compositions λ Γ by setting S λ := 0, T st (λ) =, and T st (λ, µ) = if λ is not a partition. Recall that the weak Bruhat order on W is defined by u v if and only if there exists a reduced expression v = s 1 s n for v with s 1,..., s n S and k {0,..., n} such that u = s 1 s k. Given λ Λ, [DJ, Lemma 1.5] states that D st λ := {d W d w λ } T st (λ), d t λ d. (9) is a bijection. Thus, D st λ D λ. Note that for d W one has t λ d = (t λ w λ ) d = (t λ w λ d) and therefore, d w λ t λ d T st (λ ) t λ w λ d T st (λ) w λ d D st λ. (10) By [DJ, Theorem 5.6], for given λ Λ, the elements z λ T d with w λ d W form an R-basis of S λ. By the above, this set has cardinality T st (λ ) = T st (λ), since t T(λ) is standard if and only if t T(λ ) is standard. Moreover, [DJ, Lemma 5.1] states that if d w λ and if one expands the basis element z λ T d of S λ in terms of the standard basis of M λ, i.e., z λ T d = e D λ α d,e x λ T e, with α d,e R, then and α d,wλ d = q l(d) (11) α d,e 0 ( e = w λ d or l(e) > l(w λ d) ). (12) Thus, if we set D ns λ := D λ D st λ then the R-span of the elements x λ T e, e D ns λ, is a direct complement of S λ in M λ, and we state for later reference: For every λ Γ, the Specht module S λ is R-free of rank T st (λ) and S λ has an R-complement in M λ. 7 (13)

8 2 Ascending Generalized Tableaux and Homomorphisms In this section we introduce, for any compositions λ, µ Γ, an R-submodule Hom H(M µ, M λ ) of Hom H (M µ, M λ ). In Proposition 2.2 we show that for ϕ Hom H(M µ, M λ ) and ψ Hom H(M ν, M µ ) one has ϕ ψ Hom H(M ν, M λ ). 2.1 Let λ, µ Γ be compositions of r. We say that T T(λ, µ) is ascending if, for every i N, the i-th row of T contains only entries which are greater than or equal to i. As in [BH], we denote the set of ascending and rowsemistandard elements of T(λ, µ) by T (λ, µ). One has T (λ, µ) µ λ T λ µ T (λ, µ) (14) and T (λ, λ) = {Tλ λ }. We define D λ,µ as the image of T (λ, µ) under the canonical bijection in (8). Moreover, we define By (14) we have Hom H(M µ, M λ ) := d D λ,µ Rϕ λ,µ d Hom H (M µ, M λ ). (15) Hom H(M µ, M λ ) {0} µ λ. For i N, we define the intervals P i := {λ 1 + +λ i 1 +1,..., λ 1 + +λ i } and Q i := {µ µ i 1 + 1,..., µ µ i }. One has for any w W : wt λ µ is ascending P i w Q i Q i+1 for all i 1 (P i P i+1 )w Q i Q i+1 for all i 1. We define W λ,µ to be the set of elements w W satisfying the above equivalent properties. Clearly, W λ,µ is a union of double cosets in W λ \W/W µ. The following properties are also immediate from the definition: W λ,µ if and only if µ λ, W λ,λ = W λ, W λ,µ W µ,ν W λ,ν if ν µ λ, W λ 1W µ W λ,µ if µ λ, W λ,µ W µ,ν W λ,ν if ν µ λ. (16) 8

9 The goal of this section is the proof of the following proposition. 2.2 Proposition Let λ, µ, ν Γ be compositions of r and let α Hom H(M ν, M µ ) and β Hom H(M µ, M λ ). Then β α Hom H(M ν, M λ ). Before we can prove the proposition, we need three lemmas. 2.3 Lemma Let λ, µ Γ be compositions of r and let w W λ,µ (in particular, µ λ). If w = sv with s S, v W and l(w) = l(v) + 1 then there exists ρ Γ such that µ ρ λ, s W λ,ρ and v W ρ,µ. Proof Write s = (i, i+1) with i {1,..., r 1} and note that iw > (i+1)w, since l(w) > l(sw). Let P 1, P 2,... and Q 1, Q 2,... be the subsets of {1,..., r} associated to λ and µ, respectively, as above. If there exists j N with {i, i + 1} P j then s W λ and ρ := λ satisfies the required conditions, since W λ = W λ,λ and v = sw W λ W λ,µ = W λ,µ. Hence, we may assume that there exist positive integers j < k such that i P j and i + 1 P k (we need to allow the possibility that P j+1 = = P k 1 = ). Set P l := P l for l / {j, k}, and set P j := P j {i} and P k := P k {i}. Moreover, let ρ Γ be the composition defined by P 1, P 2,.... Then clearly, ρ λ and s W λ,ρ. We still need to show that v W ρ,µ. This also implies µ ρ by (16). For l / {j+1,..., k} we have P l P l+1 = P l P l+1 and (P l P l+1 )v = (P l P l+1 )v = (P l P l+1 )sv = (P l P l+1 )w Q l Q l+1. For l {j + 1,..., k} we have i + 1 P l P l+1 = (P l P l+1 ) {i} and (P l P l+1 )v = ({i} P l P l+1 )v = ({i} P l P l+1 )sv = ({i} P l P l+1 )w {iw} Q l Q l+1, since w W λ,µ. It suffices to show that iw Q l Q l+1. But since w W λ,µ, we have (i + 1)w (P k P k+1 )w Q k Q k+1 Q l Q l+1, and iw > (i + 1)w now implies also iw Q l Q l+1. This completes the proof of the lemma. 2.4 Lemma Let λ, µ Γ and let w W λ,µ. If u W satisfies u w then also u W λ,µ. Proof We proceed by induction on l(w). Note that µ λ, since W λ,µ. If w = 1 we have u = 1 and u = w W λ,µ. Now assume that l(w) 1 and write w = sv with l(w) = l(v) + 1 and s S. By Lemma 2.3 there exists ρ Γ such that µ ρ λ, s W λ,ρ and v W ρ,µ. Since u w = sv and l(w) > l(v), we have u v or su v. If u v then l(u) l(v) < l(w) 9

10 and, by induction, v W ρ,µ implies u W ρ,µ. But W ρ,µ W λ,µ by (16). If su v then, again by induction, v W ρ,µ implies su W ρ,µ, and further, u = s(su) W λ,ρ W ρ,µ W λ,µ. Now the proof is complete. 2.5 Lemma Let λ, µ, ν Γ, v W λ,µ, and w W µ,ν. Then T v T w u W λ,ν RT u. Proof This follows immediately from Equation (1), Lemma 2.4 and W λ,µ W µ,ν W λ,ν. Proof of Proposition 2.2 By the definition in (15) we may assume that α = ϕ µ,ν d and β = ϕ λ,µ e for some d D µ,ν and e D λ,µ. By Equation (7) and since d W µ,ν and e W λ,µ, we have ϕ µ,ν d (x ν) x µ w W µ,ν RT w and (x µ ) v W λ,µ RT v. Together with Lemma 2.5 this implies ϕ λ,µ e (ϕ λ,µ e ϕ µ,ν d )(x ν) ϕ λ,µ e (x µ ) RT u = u W λ,ν w W µ,ν RT w f D λ,ν v W λ,µ w Wµ,ν u W λ fw ν RT u. RT v T w Since the elements ϕ λ,ν f, f D λ,ν, form an R-basis of Hom H (M ν, M λ ) we can write ϕ λ,µ e ϕ µ,ν d = f D λ,ν a f ϕ λ,ν f with a f R, for f D λ,ν. Thus, (ϕ λ,µ e ϕ µ,ν d )(x ν) = f D λ,ν a f ϕ λ,ν f (x ν ) = f D λ,ν a f u W λ fw ν T u. Comparing this with the above yields a f = 0 for all f D λ,ν D λ,ν, and the proof is complete. 3 The Chain Complex C λ Throughout this section we fix a composition λ Γ. We will use the R- modules Hom H(M µ, M λ ) from Section 2 and the result from Proposition 2.2 to construct a chain complex C λ of left H-modules. In the case q = 1, this chain complex coincides with the chain complex constructed in [BH] for the symmetric group algebra. 10

11 3.1 The definition of C λ. For every strictly ascending chain of length n in Γ we set γ = (λ (0) λ (n) ) (17) M γ := Hom H(M λ(0), M λ(1) ) Hom H(M λ(1), M λ(2) ) We view M γ as left H-module via Hom H(M λ(n 1), M λ(n) ) Hom(M λ(n), R). h(ϕ 1 ϕ n ε) := ϕ 1 ϕ n hε, for h H, ϕ i Hom H(M λ(i 1), M λ(i) ), i = 1,..., n, and ε Hom(M λ(n), R), where (hε)(m) := ε(mh) for m M λ(n). For every integer n 0, we write λ n for the set of chains γ as in (17) with λ (0) = λ. Further, we define the left H-module C λ n := γ λ n M γ. For n 1, γ λ n and i {1,..., n}, we denote by γ i λ n 1 the chain obtained from γ by omitting λ (i). We define an H-module homomorphism by d λ n,i : M γ M γi ϕ 1 ϕ n ε ϕ 1 ϕ i+1 ϕ i ϕ n ε, where ϕ i+1 is interpreted as ε if i = n. The direct sum of these homomorphisms yields an H-module homomorphism d λ n,i : C λ n C λ n 1 and we set n d λ n := ( 1) i 1 d λ n,i : Cn λ Cn 1 λ. i=1 Since the maps d λ n,i : Cn λ Cn 1 λ satisfy the usual simplicial relations, we obtain d λ n d λ n+1 = 0 for n 1. Thus, we have constructed a chain complex C λ : 0 C λ a(λ) d λ a(λ) Ca(λ) 1 λ d λ a(λ) 1 d λ 1 C λ

12 of finitely generated left H-modules and H-module homomorphisms. Here, a(λ) is defined as the length of the longest possible strictly ascending chain γ as in (17) with λ (0) = λ, in other words, a(λ) is the largest integer n with λ n. Note that the definition of d λ n implies immediately that d λ(0) n (ϕ 1 ϕ n ε) = (ϕ 2 ϕ 1 ) ϕ n ε ϕ 1 d λ(1) n 1(ϕ 2 ϕ n ε), (18) for n 2, γ as in (17), ϕ i Hom H(M λ(i 1), M λ(i) ), i = 1,..., n, and ε Hom(M λ(n), R). Finally, we extend the chain complex C λ by the H-module homomorphism d λ 0 : C0 λ = Hom(M λ, R) Hom(S λ, R) =: C 1 λ, ε ε S λ, and obtain a chain complex C λ : 0 C λ a(λ) d λ a(λ) Ca(λ) 1 λ d λ a(λ) 1 d λ 1 C λ 0 d λ 0 C λ 1 0 in the category of finitely generated left H-modules. In fact, in the next proposition we show that d λ 0 d λ 1 = 0. Note that C 1 λ = 0 if λ is not a partition. Also note that every H-module in the chain complex C λ is finitely generated and free as R-module so that after applying the functor Hom(, R) we obtain a chain complex starting with 0 S λ M λ and involving direct sums of modules of the form M µ with µ λ from there on. 3.2 Proposition With the notation from 3.1 one has d λ 0 d λ 1 = 0. Proof We may assume that λ is a partition, since otherwise C 1 λ = 0. Let λ µ be a chain of length 1 in Γ and let d D µ,λ. It suffices to show that (Sλ ) = 0. Since S λ = z λ H, it suffices to show that ϕ µ,λ (z λ) = 0. But, ϕ µ,λ d ϕ µ,λ d (z λ) = ϕ µ,λ d (x λt wλ y λ ) = ϕ µ,λ d (x λ)t wλ y λ x µ Hy λ and it suffices to show that x µ T w y λ = 0 for all w W. So assume that x µ T w y λ 0 for some w W. Then [DJ, Lemma 4.1] implies λ µ, with respect to relation defined in (3). But λ µ is equivalent to λ µ. Now λ = λ and λ µ implies µ λ. Since λ is a partition, we have 12 d

13 λ = λ and obtain µ µ λ, a contradiction to λ µ. This shows that x µ T w y λ = 0 for all w W and the proof is complete. 3.3 For λ Γ and an integer n 0 we denote by B λ n the set of symbols (λ (0) T1 λ (1) T2 Tn λ (n), t) (19) with γ = (λ (0) λ (n) ) λ n (so λ (0) = λ) and T i T (λ (i), λ (i 1) ) for i = 1,..., n, and with t T rs (λ (n) ). By the definition of C λ n, the elements of B λ n parametrize and R-basis of C λ n, by associating the symbol in (19) with the element ϕ λ(1),λ (0) T 1 ϕ λ(n),λ (n 1) T n ε t M γ C λ n. Note that B λ 0 = T rs (λ). For completeness we set B λ 1 := T st (λ) and associate to a standard tableau t T st (λ) the element ψ t Hom(S λ, R) with the property ψ t (z λ T d ) = 1 if d D st λ corresponds to t under the bijection (9) and ψ t (z λ T d ) = 0 otherwise. 4 Exactness Results The goal of this section is to state three exactness results, Theorems 4.2, 4.4 and 4.5, on the chain complex C λ. The lengthy and technical proof of a key lemma, Lemma 4.3, which is needed in the proof of the two latter theorems, will be postponed to the next section. The strategy of the proofs is the same as in [BH]. The proofs in this section are adaptations of the proofs in [BH]. We will present them for the reader s convenience in the Hecke algebra setting. However, the proof of Lemma 4.3 is more difficult if q Assume that λ Γ, that n 1 is an integer and that K λ n is an R-submodule of C λ n satisfying im(d λ n+1) + K λ n = C λ n (A λ n) and ker(d λ n) K λ n = {0}. (B λ n) 13

14 Then it is easy to see that the chain complex C λ is exact in degree n. In fact, the conditions (A λ n) and (Bn) λ together are equivalent to im(d λ n+1) = ker(d λ n) and ker(d λ n) K λ n = C λ n. We will produce, for certain choices of λ and n, submodules K λ n which satisfy (A λ n) and (B λ n), by taking the R-span of the basis elements of C λ n parametrized by a subset K λ n of the canonical R-basis B λ n, cf. Subsection 3.3. By abuse of notation we then will say that K λ n satisfies (A λ n) or (B λ n), if its R-span does. 4.2 Theorem The chain complex C λ is exact in degree 1 for every composition λ Γ. Proof Since S λ has an R-complement in M λ, cf. (13), the restriction map Hom(M λ, R) Hom(S λ, R) is surjective, and the proof is complete. The following lemma will be a key ingredient to the proof of Theorem 4.4, which in turn forms the base case for the inductive proof of Theorem 4.5. To state the lemma we need to introduce the following notation. For λ Γ, every tableau t T(λ) can be thought of as a sequence of numbers by appending the second row at the end of the first, and so on. On these sequences we have the lexicographic total order. We set t < t if and only if the first entry in the sequence of t which differs from the corresponding entry of t is greater than the one for t. Moreover, for t T rs (λ) we denote by C0,<t λ the R-span of the canonical basis elements ε t of Hom(M λ, R) with t > t T rs (λ). See the end of Subsection 1.2 for the definition of ε t, t T rs (λ). 4.3 Lemma Let λ Γ be a composition and let t T rs (λ). If t is not standard then ε t im(d λ 1) + C λ 0,<t. The proof of Lemma 4.3 will be given in the next section. 4.4 Theorem Let λ Γ be a composition and set K λ 0 := {(λ, t) B λ 0 t T st (λ)}. Then (A λ 0) and (B0 λ ) are satisfied. In particular, C λ every composition λ Γ. is exact in degree 0 for 14

15 Proof In order to show that (A λ 0) holds for K λ 0, we order the elements in T rs (λ) in ascending lexicographic order: t 1 < t 2 < < t n. For every i {1,..., n}, Lemma 4.3 implies that t i T st (λ) or ε ti im(d λ 1)+C0,<t λ i. An easy induction argument on i {1,..., n} now shows that ε ti im(d λ i )+K0 λ, and (A λ 0) holds. If λ is not a partition then (B0 λ ) holds trivially. Suppose that λ is a partition. In order to show that (B0 λ ) holds for K λ 0, we number the standard λ-tableaux t 1,..., t m in such a way that the following holds for i, j {1,..., m}: If t i = t λ w λ d i with unique w λ d i W, cf. (9) and (10), then l(w λ d i ) < l(w λ d j ) implies i > j. Next assume that there exists 0 ε K0 λ ker(d λ 0) and write ε = m i=1 a i ε ti with a i R. Let i be minimal with a i 0. Then, since 0 = d λ 0(ε) = ε S λ, we have m 0 = ε(z λ T di ) = a j ε tj (z λ T di ). j=i By (11) we obtain ε ti (z λ T di ) = q l(d i). Moreover, by (12) we obtain ε tj (z λ T di ) = 0 for j > i. This implies 0 = a i q l(d i), and since q is a unit, we have a i = 0, a contradiction. Thus, (B λ 0 ) holds and the proof is complete. We recall from [BH] that a composition µ Γ is called a quasi-partition if µ = µ. Here, µ is the unique smallest partition in Λ which dominates µ (cf. [BH, Lemma 4.2]) and µ Λ denotes the unique partition obtained by reordering the parts µ 1, µ 2,... of µ in weakly descending order. Recall also that a composition λ Γ is called tame if every composition µ Γ with λ µ is a quasi-partition. Tame compositions are classified in [BH, Proposition 4.6]. They include all partitions of the form (λ 1, λ 2, λ 3 ) with λ 3 {0, 1}. The technical definition of a tame composition is dictated by the inductive proof of the following theorem. The proof is a straightforward adaptation of the proof of Theorem 5.1 in [BH] and will not be repeated here. 4.5 Theorem Let λ Γ be a tame composition. For n = 1 set K λ 1 :=, for n = 0 set K λ 0 := T st (λ) and for n 1 set K λ n := { (λ = λ (0) T1 λ (1) T2 Tn λ (n), t) B λ n t T st (λ (n)}. Then (A λ n) and (B λ n) hold for all n 1. In particular, the chain complex C λ is exact. 15

16 5 Proof of Lemma 4.3 The exclusive goal of this section is to prove Lemma 4.3. The proof is a substantial refinement of the proof of [BH, Lemma 3.4]. Throughout we fix a composition λ Γ and a λ-tableau t T rs (λ) T st (λ). We first need to establish some notation that will be used throughout this section. 5.1 Since t T rs (λ) is not standard, there exist consecutive rows of t, say rows k and k + 1, which we write as and a 1 a 2 a m (m = λ k ) b 1 b 2 b n (n = λ k+1 ), such that there exist i {0,..., min{m, n 1}} satisfying a 1 < b 1, a 2 < b 2,, a i < b i, and (a i+1 > b i+1 or m = i). The case m = i can arise when λ k+1 > λ k. We also set Z := {a 1,..., a m, b 1,..., b n }. For every subset X Z with X n we set µ X := (λ 1,..., λ k 1, λ k + n X, X, λ k+2,...) Γ and denote by t X T rs (µ X ) the tableau which coincides with t in all rows except row k and row k+1, and which has the elements of X in row k+1 and those of ZX in row k. Moreover, we denote by d X D µx the element which satisfies t X = t µ X d X, and by ε X Hom(M µ X, R) the element ε tx = ε dx. We will also use the composition We then have ν X := (λ 1,..., λ k, n X, X, λ k+2,...) Γ. ν X λ µ X and W νx = W µx W λ. Finally, we define the adjacent, pairwise disjoint intervals of integers, A X := {λ λ k 1 + 1, λ λ k 1 + 2,..., λ λ k 1 + λ k }, B X := {λ λ k + 1,..., λ λ k + (n X )}, C X := {λ λ k + (n X + 1),..., λ λ k + n}, 16

17 (with n = λ k+1 ) and set D := {1,..., r} (A X B X C X ) Thus, (A X, B X, C X ) are the k-th, (k + 1)-th, (k + 2)-th respective subsets of the set partition associated with ν X, (A X, B X C X ) are the k-th and (k+1)-th respective subsets of the set partition associated with λ, and (A X B X, C X ) are the k-th and (k + 1)-th subsets associated with µ X. Recall from Subsection 1.3 that, for X as above, one has an element ϕ µ X,λ 1 Hom H(M λ, M µ X ). 5.2 Lemma Let X Z be a subset with X n. (a) For d D λ, the set W X,d := W µx (D νx W λ )dd 1 X has at most one element. (b) In Hom(M λ, R), one has the equation ε X ϕ µ X,λ 1 = q l(wx,d) ε d, d D λ W X,d where we write W X,d = {w X,d } when W X,d. Proof (a) Let d D λ and assume that one has elements w 1, w 2 W µx and e 1, e 2 D νx W λ with w i = e i dd 1 X for i = 1, 2. Then e 1 e 1 2 = w 1 w2 1 W µx W λ = W νx. Since e 1, e 2 D νx, we have e 1 = e 2 and w 1 = w 2. (b) Recall that the elements x λ T d, d D λ, form an R-basis of M λ. For d D λ we have ( ε X ϕ µ X,λ 1 (x λ T d ) ) ( ) = ε X ϕ µ X,λ 1 (x λ )T d = ε X (x µx T ed ) e D νx W λ by (7). For every e D νx W λ there exists a unique f d,e D µx with ed W µx f e,d and a unique element w e,d W µx with ed = w e,d f e,d. We continue the above computation: ( ε X ϕ µ X,λ 1 (x λ T d ) ) = ε X (x µx T we,d f e,d ) e D νx W λ = q l(w e,d) ε X (x µx T fe,d ) = e D νx W λ 17 e Dν X W λ f e,d =d X l(w q e,d ).

18 But, if e D νx W λ satisfies f e,d = d X then w e,d = edd 1 X W X,d = {w X,d } and e is the only element in D νx W λ with f e,d = d X. Conversely, if W X,d and we write w X,d = edd 1 X with e D νx W λ then e satisfies f e,d = d X. Thus, we have ε X ( ϕ µ X,λ 1 (x λ T d ) ) = 0 if W X,d =, q X,d) if W X,d. Now the statement in (b) is immediate and the proof is complete. 5.3 Lemma Let X Z be a subset with X n. (a) The function {X Y Z Y = n} D λ, Y d Y, is injective with image {d D λ W X,d }. (b) In Hom(M λ, R), one has ε X ϕ µ X,λ 1 = q l(w X,Y ) ε Y, X Y Z Y =n where w X,Y := w X,dY. (c) For X Y Z with Y = n one has l(w X,Y ) = {z Z Y y < z}. y Y X Proof (a) For X Y Z with Y = n we have d Y D λ, since µ Y = λ. Moreover, the map Y d Y is clearly injective. Our next goal is to show that W X,dY. Note that X Y = (B X C X )d Y implies (X)d 1 Y B X C X. We define an element e W by e AX D is the identity, e CX : C X (X)d 1 Y is monotonous, e BX : B X (B X C X ) (X)d 1 is monotonous. Y (20) Then, e D νx, since e is monotonous on A X B X, on C X and on D. Moreover, e W λ, since (B X C X )e = B X C X and e is the identity on A X D. We set w := ed Y d 1 X. (21) 18

19 We claim that In fact, for x D we have (x)ed Y d 1 X coincide on D. Also, w CX w CX D is the identity map. (22) is the composition = (x)d Y d 1 X = x, since d X and d Y C e X (X)d 1 Y d Y X d 1 X C X. This composition is monotonous, since each of the three factors is monotonous (because e D νx, d Y D λ, d X D µx and (C X )e B X C X ). Thus, w CX is the identity map and Claim (22) is proven. Since w is the identity on D C X, we obtain (A X B X )w = A X B X. This implies w W µx and w W X,dY. Conversely, let d D λ with W X,d. Then there exists w W µx and e D νx W λ with w = edd 1 X. Since w W µx, we have (C X )w = C X, and since e W λ, we have (C X )e (B X C X )e = B X C X. Therefore, Y := (B X C X )d (C X )ed = (C X )wd X = (C X )d X = X and Y = B X C X = n. We claim that d = d Y. For j / {k, k + 1} and D j := {λ λ j 1 + 1,..., λ λ j 1 + λ j } we have (D j )d = (D j )ed = (D j )wd X = (D j )d X = (D j )d Y, since e W λ and w W µx. Moreover, d and d Y are monotonous on D j, since d, d Y D λ. Therefore, d and d Y coincide on D j. Further, (B X C X )d Y = Y = (B X C X )d and d and d Y are monotonous on B X C X. Therefore, d and d Y coincide on B X C X. The above implies that (A X )d = (A X )d Y and since d and d Y are monotonous on A X, they also coincide on A X. Thus, d = d Y. (b) Using Lemma 5.2(b) and part (a), we have ε X ϕ µ X,λ 1 = d D λ W X,d q l(w X,d) ε d = X Y Z Y =n q l(w X,d Y ) ε dy and (b) is proven. (c) By (22), w X,Y is the identity on D C X, and we can view w X,Y as a permutation of A X B X. We claim that w := w X,Y is the permutation of A X B X with maps B X monotonously onto the positions of Y X in the 19

20 k-th row of t X. In fact, we need to show that (B X )wd X = Y X and that w is monotonous on B X. But by (20) we have (B X )wd X = (B X )ed Y = ( ) (B X C X ) (X)d 1 Y dy = Y X, where e is defined as in Part (a). Moreover, since (B X )w A X B X and since d X is monotonous on A X B X, w is monotonous on B X if and only if wd X = ed Y is. Again, since (B X )e B X C X and d Y is monotonous on B X C X, ed Y is monotonous on B X if and only if e is. But, e D µx is monotonous on A X B X and so also on B X. This shows the claim. Since l(w) is equal to the number of inversions of w, where w is viewed as a permutation of A X B X, our claim immediately implies the desired equation if we can also show that w is monotonous on A X. But, w is monotonous on A X if and only if wd X = ed Y is, by the argument above. Further, e is the identity on A X, cf. (20), so that with d Y also ed Y is monotonous on A X, and the proof of the lemma is complete. where and Next we consider the element c 1 := ( 1) X q f(x)(,λ µ X ϕ 1 ε X) C λ 1, X {a 1,...,a i } X := X {b i+2,, b n } f(x) := j {1,...,i} a j X (m + 1 j). 5.4 Lemma With the above notation, one has for some integer l. d λ 1(c 1 ) q l ε t + C 0,<t, Proof By Lemma 5.3(b) we have d λ 1(c 1 ) = ( 1) X q f(x)( ε X ϕ = = X {a 1,...,a i } X {a 1,...,a i } {b i+2,...,bn} Y Z Y =n ( 1) X q f(x) ( X Y Z Y =n X {a 1,...,a i } Y 20,λ µ X 1 ) q l(w X,Y ) ε Y ( 1) X q f(x)+l(w X,Y )) ε Y.

21 For every Y Z with {b i+2,..., b n } Y and Y = n we study the contribution of the corresponding summand at the end of the last equation. We distinguish three cases. If Y = {b 1,..., b n } then ε Y = ε t and the only possible set X in the inner sum of the last equation is the empty set. Therefore, the summand corresponding to Y is equal to q l ε t with l = f( ) + ). l(w,y If Y {a 1,..., a i } = and Y {b 1,..., b n } we claim that t Y < t. Note that in this case we have m 1. Let j {1,..., i+1} be minimal with b j / Y (note that a 1 < b j ) and let p {j 1,..., i} be maximal with a p < b j (note that a j 1 < b j 1 < b j ). Then the k-th row of t Y begins with a 1, a 2,..., a p, b j and b j < a p+1. Note that p < m, since if p = m then a 1,..., a m Z Y and therefore Y {b 1,..., b n }, a contradiction. Now our claim is proven and the contribution of the summand corresponding to Y is an element in C0,<t. λ We are left with the case that Y {a 1,..., a i }. If t Y < t we are done. So assume that t Y > t. Our goal is to show that ( 1) X q f(x)+l(w X,Y ) = 0. X {a 1,...,a i } Y First let j {1,..., i} be minimal with a j Y. We claim that {z Z Y z < a j } = {a 1,..., a j 1 }. (23) Clearly, the right-hand side is contained in the left-hand side. Now let z Z Y with z < a j and assume that z is not contained in the right-hand side. Then z = b p for some p {1,..., i + 1}, since b i+2,..., b n Y. Let r {1,..., i + 1} be minimal with b r Z Y and b r < a j, and let s {1,..., j 1} be maximal with a s < b r (note that j > 1, since b r < a j and that a 1 < b r ). Then the k-th row of t Y starts with a 1, a 2,..., a s, b r and one has b r < a s+1. This implies t Y < t, a contradiction. Therefore, the claim (23) is proven. For our given set Y, the subsets X of {a 1,..., a i } Y = {a j,..., a i } Y fall into two classes, the ones that contain a j and the ones that don t. So let X 1 be a subset of {a 1,..., a i } Y with a j X 1 and let X 2 := X 1 {a j }. Since X 1 X 2 defines a bijection between these two classes, it suffices to show that or, equivalently, that ( 1) X 1 q f(x 1)+l(w X 1,Y ) + ( 1) X 2 q f(x 2)+l(w X 2,Y ) = 0, l(w X1,Y ) + m + 1 j = l(w X2,Y ). (24) 21

22 But by Lemma 5.3(c) and Equation (23), and since Y X 2 = Y X 1 {a j } and Z Y = m, we have l(w X2,Y ) l(w X1,Y ) = y Y X2 {z Z Y y < z} y Y X1 {z Z Y y < z} = {z Z Y a j < z} = m {z Z Y z < a j } = m (j 1). This shows Equation (24) and the proof of the lemma is complete. Now Lemma 4.3 is an immediate consequence of Lemma 5.4. References [A] [AB1] [AB2] [BH] [DJ] [Dn] K. Akin: On complexes relating the Jacobi-Trudi identity with the Bernstein-Gelfand-Gelfand resolution. J. Algebra 117 (1988), K. Akin, D. A. Buchsbaum: Characteristic-free representation theory of the general linear group. Adv. in Math. 58 (1985), K. Akin, D. A. Buchsbaum: Characteristic-free representation theory of the general linear group II. Homological considerations. Adv. in Math. 72 (1988), R. Boltje, R. Hartmann: Permutation resolutions for Specht modules. J. Algebraic Combin. 34 (2011), R. Dipper, G. James: Representations of Hecke algebras of general linear groups. Proc. London Math. Soc. 52 (1986), S. Donkin: Finite resolutions of modules for reductive algebraic groups. J. Algebra 101 (1986), [Dy] S. R. Doty: Resolutions of B modules. Indag. Mathem. 5(3) (1994), [S] A. P. Santana: The Schur algebra S(B + ) and projective resolutions of Weyl modules. J. Algebra 161 (1993),

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