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ROW AND COLUMN REMOVAL THEOREMS FOR HOMOMORPHISMS OF SPECHT MODULES AND WEYL MODULES ´ SINEAD LYLE AND ANDREW MATHAS Abstract. We prove a q-analogue of the row and column removal theorems for homomor- phisms between Specht modules proved by Fayers and the ﬁrst author [6]. The original paper considered a complement to (Donkin’s generalisations of) James’s row and column removal theorems regarding Specht modules for the symmetric groups. Here, we consider the Specht modules of the Iwahori–Hecke algebra HR,q (Sn ) when q = −1. Similar theorems are then shown to hold in the context of homomorphisms between the Weyl modules of the q-Schur algebra. In fact, the latter results are shown to hold in more generality: speciﬁcally we can allow q = −1. 1. Introduction Fix a positive integer n ≥ 1 and a commutative ring R. Let q be an invertible element of R and let H = Hn = HR,q (Sn ) be the Iwahori–Hecke algebra of the symmetric group S n of degree n. For each partition λ of n Dipper and James [2] deﬁned a right H R,q (Sn )-module S λ called a Specht module. These modules are important because when H R,q (Sn ) is semisimple the Specht modules form a complete set of pairwise non–isomorphic irreducible H R,q (Sn )-modules. More generally, every irreducible HR,q (Sn )-module arises as the head of some Specht module. In this paper we prove the following two theorems, which are q-analogues of the row and column removal theorems for homomorphisms from [6]. Theorem 1.1. Let R be a ring and q an invertible element of R with q = −1. Let λ and µ be partitions of n, and suppose that for some s we have λ1 + · · · + λ s = µ 1 + · · · + µ s . Deﬁne partitions µT = (µ1 , . . . , µs ), λT = (λ1 , . . . , λs ), µB = (µs+1 , µs+2 , . . . ), λB = (λs+1 , λs+2 , . . . ). Then, putting m = |λB | = |µB |, we have T T B B HomHn (S µ , S λ ) ∼ HomHn−m (S µ , S λ ) ⊗ HomHm (S µ , S λ ). = For any partition ν, let ν be the partition which is conjugate to ν; that is, ν is the partition obtained by swapping the rows and columns of ν. Theorem 1.2. Let R be a ring and q an invertible element of R with q = −1. Let λ and µ be partitions of n, and suppose that for some r we have λ1 + · · · + λ r = µ 1 + · · · + µ r . 1991 Mathematics Subject Classiﬁcation. 20C08, 05E10. Key words and phrases. Hecke algebras, Schur algebras. This research was supported by ARC grant DP0343023. 1 2 ´ SINEAD LYLE AND ANDREW MATHAS Deﬁne λL = min(λ1 , r), min(λ2 , r), . . . , µL = min(µ1 , r), min(µ2 , r), . . . , λR = max(λ1 − r, 0), max(λ2 − r, 0), . . . , µR = max(µ1 − r, 0), max(µ2 − r, 0), . . . . Then, putting l = |λR | = |µR |, we have L L R R HomHn (S λ , S µ ) ∼ HomHn−l (S λ , S µ ) ⊗ HomHl (S λ , S µ ). = The partitions λ, λL , λR , λT , λB appearing in these theorems may be viewed pictorially as follows: λT λR λ λ L λB . One of the reasons why these results are interesting is that Donkin [4] and James [8] have proved analogous results for the decomposition numbers of H. To state these results precisely, deﬁne e > 1 to be minimal such that 1 + q + · · · + q e−1 = 0 (setting e = ∞ if no such integer exists) and if S is an H-module and D is a simple H-module let [S : D] be the decomposition multiplicity of D in S. Finally, if λ is an e-restricted partition (that is, λ i − λi+1 < e for all i), let D λ = S λ / rad S λ be the corresponding simple H-module (see, for example, [9]). Then, under the assumptions of Theorem 1.1 we have T T B B [S µ : D λ ] = [S µ : D λ ] · [S µ : D λ ]. There is an analogous statement under the assumptions of Theorem 1.2. This paper is structured as follows. In the next section we recall the necessary machinery from the literature and also prove some preliminary combinatorial results. Section 3 gives the proof of Theorem 1.1 and Theorem 1.2. Finally, in section 4, we establish analogous results relating homomorphisms between Weyl modules of the q-Schur algebras. For precise statements, see Theorem 4.1 and Theorem 4.2. 2. Definitions We begin by establishing some deﬁnitions. The notation used will be that of [9]. Deﬁnition 2.1. Let n be a positive integer and S n the symmetric group acting on n letters. Suppose R is a ring and let q be an invertible element of R. The Iwahori–Hecke algebra H = HR,q (Sn ) of Sn is the unital associative R-algebra with generators T 1 , T2 , . . . , Tn−1 and relations (Ti − q)(Ti + 1) = 0, for i = 1, 2, . . . , n − 1, Ti Tj = T j Ti , for 1 ≤ i < j − 1 ≤ n − 2, Ti+1 Ti Ti+1 = Ti Ti+1 Ti , for i = 1, 2, . . . , n − 2. Set si to be the transposition (i, i + 1) ∈ Sn . Let ω ∈ Sn and suppose si1 si2 . . . sil is a reduced expression for ω; that is, if ω = sj1 sj2 . . . sjk then k ≥ l. We say that ω has length l and write (ω) = l. Deﬁne Tω = Ti1 Ti2 . . . Tik . Then H is free as an R-module with basis {T ω | ω ∈ Sn }. Deﬁnition 2.2. Let λ and µ be compositions of n. Say that λ dominates µ and write λ µ if k k λi ≥ µi i=1 i=1 for all k. If λ µ and λ = µ, we write λ µ. ROW AND COLUMN REMOVAL THEOREMS 3 Deﬁnition 2.3. Let λ = (λ1 , λ2 , . . .) be a composition of some integer n. The diagram of λ is the set {(i, j) | 1 ≤ i and 1 ≤ j ≤ λi } regarded as an array of nodes in the plane. A λ-tableau is obtained from the diagram of λ by replacing each node with one of the integers 1, 2, . . . , n, allowing no repeats. If t is a λ-tableau we write Shape(t) = λ. A λ-tableau t is row standard if its entries increase along the rows; and is standard if λ is a partition and the entries of t increase across the rows and down the columns. If λ is a partition, deﬁne Std(λ) to be the set of standard λ-tableaux. The symmetric group S n acts from the right in a natural way on the set of tableaux. Deﬁnition 2.4. Let λ and µ be compositions of n. A λ-tableau of type µ is obtained by replacing each node in the diagram of λ with one of the integers 1, 2, . . . such that the number i occurs µi times. A λ-tableau T (of type µ) is semistandard if λ is a partition, and the entries of T are non-decreasing across the rows and strictly increasing down the columns. Let T 0 (λ, µ) denote the set of semistandard λ-tableaux of type µ, and let T λ denote the unique semistandard λ-tableau of type λ. Deﬁnition 2.5. Let λ be a composition of n. Deﬁne t λ to be the standard λ-tableau in which the integers 1, 2, . . . , n are entered in increasing order along its rows, and S λ to be the row stabilizer of tλ . Set mλ = Tω . ω∈Sλ We deﬁne the permutation module Mλ to be the right H-module M λ = mλ H. Deﬁnition 2.6. Suppose that λ is a composition of n and that t is a row standard λ-tableau. Deﬁne d(t) ∈ Sn to be such that t = tλ d(t). Now deﬁne ∗ to be the R-linear anti–automorphism of H determined by Ti∗ = Ti . If s, t are row standard λ-tableaux, deﬁne m st ∈ H by ∗ mst = Td(s) mλ Td(t) . Then Murphy [10] has shown that {mst | s, t ∈ Std(λ) for some partition λ of n} is a cellular basis of H in the sense of Graham and Lehrer [7]. Further, if µ is any composition of n and s, t are row standard µ-tableaux then m st can be written as a R-linear combination of muv where u, v ∈ Std(λ), and λ is a partition of n such that λ µ. Deﬁnition 2.7. Let λ, µ be compositions of n and let t be a λ-tableau. Deﬁne µ(t) to be the λ-tableau of type µ obtained by replacing each entry in t with its row index in t µ . Suppose that S is a semistandard λ-tableau of type µ and t is a standard λ-tableau. Deﬁne mSt = mst s ∈ Std(λ) and µ(s) = S . s As proved in [10], M µ is free as an R-module with basis {mSt | S ∈ T0 (λ, µ), t ∈ Std(λ) for some partition λ of n}. Deﬁnition 2.8. Let λ be a partition of n. Deﬁne H λ to be the two–sided ideal of H with basis {mst | s, t ∈ Std(ν) where ν λ} ˇ and Hλ to be the two–sided ideal of H with basis {mst | s, t ∈ Std(ν) where ν λ}. We deﬁne the Specht module Sλ ˇ to be the right H-module (Hλ + mλ )H. 4 ´ SINEAD LYLE AND ANDREW MATHAS ˇ It follows that S λ is free as an R-module with basis {Hλ + mtλ t | t ∈ Std(λ)}. We now look at the q-Schur algebra. Fix a non–negative integer d and let Λ(d, n) be the set of compositions of n with at most d non–zero parts. Let Λ + (d, n) be the set of partitions contained in Λ(d, n) and for λ ∈ Λ+ (d, n), set T0 (λ) = µ∈Λ(d,n) T0 (λ, µ). Deﬁnition 2.9. The q-Schur algebra S = S(d, n) is the endomorphism algebra of H deﬁned by S = EndH M µ . µ∈Λ(d,n) Deﬁnition 2.10. Let λ, µ ∈ Λ(d, n), ν ∈ Λ + (d, n) and suppose S ∈ T0 (ν, λ), T ∈ T0 (ν, µ). If τ ∈ Λ(d, n), deﬁne ϕST ∈ S by ϕST (mτ h) = δτ µ mst h s, t ∈ Std(ν) and λ(s) = S, µ(t) = T ∈ Mλ s,t for all h ∈ H. In fact [9, 4.13], {ϕST | ν ∈ Λ+ (d, n) and S, T ∈ T0 (ν)} is a cellular basis of S. Deﬁnition 2.11. Let λ ∈ Λ+ (d, n), µ ∈ Λ(d, n) and T ∈ T0 (λ, µ). Deﬁne the map ϕT : M τ → Sλ τ ∈Λ(d,n) by ϕT (mτ h) = δτ µ ˇ Hλ + mλ Td(t) h t ∈ Std(λ) and µ(t) = T t for all h ∈ H. By restriction, we will consider ϕT to be an element of HomH (M µ , S λ ). Dipper and James [3, Cor. 8.7] have proved a stronger version of the next result for the Specht modules which are dual to the S λ . This lemma can be deduced from the Dipper–James result; however, we give an independent proof of it because the lemma is central to what follows (and the Dipper–James result is more diﬃcult to prove). If s is a standard tableau let s↓m be the subtableau of s which contains the numbers 1, . . . , m. If s and t are standard λ-tableaux write s t if Shape(s↓m) Shape(t↓m), for m = 1, . . . , n. If s t and s = t we write s t. The following Lemma should be well–known, however, we do not know of a reference in the literature. (After much work, Dipper and James [3, Cor. 8.7] prove the corresponding result for Specht modules which are dual to the ones considered here.) Lemma 2.12. Let λ and µ be partitions of n and suppose that q = −1. Then Hom H (M µ , S λ ) is free as an R-module with basis {ϕT | T ∈ T0 (λ, µ)}. Proof. First observe that if φ ∈ HomH (M µ , S λ ) then φ(mµ h) = φ(mµ )h, for all h ∈ H, so φ is completely determined by the value of φ(m µ ). In particular, the maps ϕT are linearly indepen- dent because {ϕT (mµ ) | T ∈ T0 (λ, µ)} is linearly independent by the remark after Deﬁnition 2.8. It remains to show that the set {ϕT (mµ ) | T ∈ T0 (λ, µ)} spans HomH (M µ , S λ ). Let φ ∈ HomH (M µ , S λ ) and write φ(mµ ) = {Hλ +ˇ rt mtλ t | t ∈ Std(t)}, for some rt ∈ R. Fix a tableau s such that rs = 0 and rt = 0 whenever s t. We claim that µ(s) is a semistandard ROW AND COLUMN REMOVAL THEOREMS 5 tableau. If not, then there exist integers i < j which are in the same column of s such that (i, j) ∈ Sµ . Now, (j − 1, j) ∈ Sµ so mµ Tj = qmµ . Therefore (†) ˇ Hλ + ˇ rt mtλ t = φ(mµ ) = q −1 φ(mµ Tj−1 ) = q −1 φ(mµ )Tj−1 = Hλ + q −1 rt mtλ t Tj−1 . Let s = s(j − 1, j). If i < j − 1 then s is a standard tableau and s dominates s , so mtλ s Tj−1 = mtλ s , and rs = 0 by the choice of s. However, now (†) and [9, Cor. 3.9] show that m tλ s appears with non–zero coeﬃcient in φ(m µ ), so this is a contradiction and we must have i = j −1. Therefore, j − 1 and j are in the same column of s and so there exist a v ∈ R such that ˇ ˇ Hλ + mtλ s Tj−1 = Hλ + −mtλ s , + {av mtλ v | v s}, by [9, Cor. 3.21]. Hence, comparing the coeﬃcient of m tλ s on both sides of (†) shows that rs = −q −1 rs . Since q = −1 we again have a contradiction. Hence, µ(s) is semistandard as claimed. To complete the proof it is enough to show that r t = rt whenever µ(t) = µ(t ). The proof of this is similar to [9, Prop. 3.18] and is left as an exercise for the reader. Deﬁnition 2.13. Let µ ∈ Λ(d, n). Deﬁne ϕµ = ϕT µ T µ ∈ S. Then ϕµ acts as the identity on M µ and zero on M τ for τ = µ. Set Mµ to be the right S-module Mµ = ϕµ S. Then Mµ has a basis given by {ϕST | S ∈ T0 (λ, µ), T ∈ T0 (λ) where λ ∈ Λ+ (d, n)}. Deﬁnition 2.14. For λ ∈ Λ+ (d, n), we deﬁne S λ to be the two-sided ideal of S given by S λ = {ϕST | S, T ∈ T0 (ν) where ν ∈ Λ+ (d, n) and ν λ} ˇ and S λ to be the two-sided ideal of S given by ˇ S λ = {ϕST | S, T ∈ T0 (ν) where ν ∈ Λ+ (d, n) and ν λ}. We deﬁne the Weyl module Wλ to be the right S-module ˇ W λ = (S λ + ϕλ )S. Then W λ is free as an R-module with basis (2.15) ˇ {S λ + ϕT λ T | T ∈ T0 (λ)}. ˇ In fact, we may identify S λ + ϕT λ T with the map ϕT ∈ HomH ( τ M τ , S λ ) of Deﬁnition 2.11. Then W λ is isomorphic to the S-submodule of Hom ( τ λ H τ M , S ) with basis {ϕT | T ∈ T0 (λ)}. (Note that if q = −1 this is actually a basis of Hom H ( τ M τ , S λ ) by Lemma 2.12.) Deﬁnition 2.16. Let λ ∈ Λ+ (d, n), µ ∈ Λ(d, n) and T ∈ T0 (λ, µ). Deﬁne θT ∈ HomS (Mµ , W λ ) by θT (ϕµ f ) = ϕT f for all f ∈ S. Using (2.15) is now straightforward to prove the following. Lemma 2.17. Suppose that λ ∈ Λ+ (d, n) and µ ∈ Λ(d, n). Then {θT | T ∈ T0 (λ, µ)} is a basis of HomS (Mµ , W λ ). 6 ´ SINEAD LYLE AND ANDREW MATHAS 3. The Hecke algebras of type A In this section we prove Theorem 1.1 and Theorem 1.2. We start by showing that these two results are equivalent. Lemma 3.1. For any partitions λ and µ of n, we have HomHn (S µ , S λ ) ∼ HomHn (S λ , S µ ). = Proof. For any Hn -module M , let M = HomR (M, R) be the dual of M . Then there exists an # Hn -automorphism # such that (S ν )# ∼ (S ν ) [10, Theorem 5.2]; explicitly, Tω = (−q) (ω) Tω−1 = −1 for all ω ∈ Sn . Hence HomH (S µ , S λ ) ∼ HomH ((S µ )# , (S λ )# ) n = n ∼ HomH ((S µ ) , (S λ ) ) = n ∼ HomH (S λ , S µ ). = n Proposition 3.2. Theorem 1.1 holds if and only if Theorem 1.2 holds. Proof. If λ and µ satisfy the conditions of Theorem 1.1, then λ and µ satisfy the conditions of Theorem 1.2, with r = s, l = m and with (λ )L = (λT ) , (λ )R = (λB ) , (µ )L = (µT ) , (µ )R = (µB ) . By Lemma 3.1 we have HomHn (S µ , S λ ) ∼ HomHn (S λ , S µ ) and = T T B B HomHn−m (S µ ,S λ ) ⊗ HomHm (S µ , S λ ) ∼ HomH λL L R R = n−m (S , S µ ) ⊗ HomHm (S λ , S µ ). The proof of the converse is similar. It is therefore suﬃcient to prove only Theorem 1.1. Henceforth in this section, we ﬁx a ring R and an invertible element q ∈ R such that q = −1. We also ﬁx integers m and n with 0 ≤ m ≤ n and partitions λ and µ of n which satisfy λ1 + · · · + λ s = µ 1 + · · · + µ s = n − m for some s. For any composition ν of n which satisﬁes ν 1 + · · · + νs = n − m write ν T = (ν1 , . . . , νs ) and ν B = (νs+1 , νs+2 , . . .). Deﬁnition 3.3. Let H = Hn be the algebra generated by T1 , T2 , . . . , Tn−1 subject to the re- lations of Deﬁnition 2.1 The subalgebra H T ∼ Hn−m is deﬁned to be the subalgebra of H = generated by T1 , T2 , . . . , Tn−m−1 B ∼ H is deﬁned to be the subalgebra of H generated by and the subalgebra H = m Tn−m+1 , Tn−m+2 , . . . , Tn−1 . Note that ifhT ∈ HT and hB∈ HB then hT hB = hB hT . Deﬁne Sn−m to be the permutation group acting on the set {1, 2, . . . , n − m} and S B the m permutation group acting on {n − m + 1, n − m + 2, . . . , n}. Suppose that ν is any partition T B of n such that ν1 + ν2 + . . . + νs = n − m. Deﬁne tν and tν to be the standard ν-tableaux ROW AND COLUMN REMOVAL THEOREMS 7 with entries 1, 2, . . . , n − m and n − m + 1, n − m + 2, . . . , n respectively which increase along the rows, and deﬁne Sν T ⊆ Sn−m and SBB ⊆ SB accordingly. Now deﬁne ν m mν T = Tω , ω∈Sν T mν B = Tω . ω∈SBB ν Note that mν = m ν T mν B . Then we have T M ν = mν T H T , B M ν = mν B H B . Hence we have cellular bases of H T and HB using Deﬁnition 2.6 in the obvious way. We may ˇ T ˇ B T B similarly deﬁne Hν , Hν , S ν and S ν . Deﬁnition 3.4. Suppose that AT ∈ T0 (λT , µT ) has entry xi in position i of row j and AB ∈ j i T0 (λB , µB ) has entry yj in position i of row j. Deﬁne (AT , AB ) to be the λ-tableau of type µ i with entry zj in position i of row j, where i xi j (j ≤ s) zj = i . yj + s (j > s) For example, if 111112 112 AT = 2223 , AB = 23 , 333 3 then 1 1 1112 2 2 23 T B 3 3 3 (A , A ) = 4 4 5 . 5 6 6 Lemma 3.5. If AT and AB are semistandard, then so is (AT , AB ), and {(AT , AB ) | AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB )} is precisely the set of semistandard λ-tableaux of type µ. Proof. The ﬁrst statement is clear. Suppose A is a semistandard λ-tableau of type µ with entry i zj in position i of row j. Then the entries of A strictly increase down the columns, so that for i j > s we have that zj > s. Since the number, n − m, of entries less than or equal to s equals the i number of positions in rows 1, . . . , s, it also is also true that z j ≤ s for j s. So the top s rows of A constitute a λ T -tableau of type µT (which is clearly semistandard), and a similar statement holds for the lower rows. Deﬁnition 3.6. Suppose that a is a ν-tableau which contains the entries 1, 2, . . . , n − m in rows 1, 2, . . . , s. Deﬁne aT to be the ν T -tableau consisting of rows 1 to s of a, and a B the ν B -tableau consisting of rows s + 1, s + 2, . . . Similarly, if we have such an a T , aB , deﬁne a = (aT , aB ) in the obvious manner. Say that aT ∈ Std(ν T ) if it is a standard ν T -tableau containing the entries 1, 2, . . . n − m and aB ∈ Std(ν B ) if it is a standard ν B -tableau containing the entries n − m + 1, n − m + 2, . . . n. 8 ´ SINEAD LYLE AND ANDREW MATHAS Lemma 3.7. Let A ∈ T0 (ν, µ). Then a | a ∈Std(ν) and µ(a) = A = (aT , aB ) | aT ∈ Std(ν T ) and µT (aT ) = AT , aB ∈ Std(ν B ) and µB (aB ) = AB . Proof. This follows from Lemma 3.5. T T B B Deﬁnition 3.8. Now suppose that ΦT ∈ HomHT (M µ , S λ ), and ΦB ∈ HomHB (M µ , S λ ). Then ΦT = ΓAT ϕAT AT ∈ T0 (λT , µT ) AT and ΦB = ΓAB ϕAB AB ∈ T0 (λB , µB ) AB for some ΓAT , ΓAB ∈ R. Deﬁne Φ = (ΦT , ΦB ) ∈ HomH (M µ , S λ ) by Φ= ΓAT ΓAB ϕ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB ) . AT AB By Lemma 2.12 and Lemma 3.5 this deﬁnes an R-linear bijection T T B B HomHT (M µ , S λ ) ⊗ HomHB (M µ , S λ ) → HomH (M µ , S λ ). Lemma 3.9. Suppose that λ ν µ. Then ν 1 + . . . + νs = n − m and either ν T µT or ν B µB . Proof. The proof follows directly from the deﬁnition of . Lemma 3.10. Suppose that ν µ and that S ∈ T 0 (ν, µ). Let ∗ m= Td(s) mν s ∈ Std(ν) and µ(s) = S . s ˇ T Then m = mµ hT hB for some hT , hB such that hT ∈ HT and hB ∈ HB and either mµT hT ∈ Hµ ˇ B or mµB hB ∈ Hµ . Proof. Deﬁne mT = Td(sT ) mν T | sT ∈ Std(ν T ) and µT (sT ) = S T . ∗ sT Then mT ∈ M µT and so mT = mµT hT for some hT ∈ HT . Similarly, we can deﬁne mB = Td(sB ) mν B sB ∈ Std(ν B ) and µB (sB ) = S B . ∗ sB so that mB = mµB hB for some hB ∈ HB . If ν T ˇ T µT then mµT hT ∈ Hµ ; otherwise, by Lemma ˇ B 3.9, ν B µB and mµB hB ∈ Hµ . Then ∗ m= Td(s) mν s ∈ Std(ν) and µ(s) = S s = Td(sT ) Td(sB ) mν T mν B sT ∈ Std(ν T ) and µT (sT ) = S T , ∗ ∗ sT sB sB ∈ Std(ν B ) and µB (sB ) = S B ROW AND COLUMN REMOVAL THEOREMS 9 by Lemma 3.7 = m T mB = m µT h T m µB h B = m µ hT hB . Proposition 3.11. Suppose that λ, µ, ν are partitions of n such that λ, ν µ and λ ν. Let S ∈ T0 (ν, µ), t ∈ Std(ν) and A ∈ T0 (λ, µ). Then ϕA (mSt ) = 0. Proof. Consider the map ϕT λ A of Deﬁnition 2.10. Then ϕA = πϕT λ A where π : M λ → S λ is the natural projection, and ϕT λ A consists of left multiplication by certain elements of H. Now, since mSt ∈ Hν , we have that ϕT λ A (mSt ) ∈ Hν , and hence can be written as an R-linear combination of elements of the form muv where u, v ∈ Std(τ ) for some τ ν. However, ϕT λ A (mSt ) ∈ M λ , and hence can be written as an R-linear combination of elements of the form muv where u, v ∈ Std(τ ) for some τ λ. Since λ ν, this shows that ϕ T λ A (mSt ) ∈ Hλˇ and so ϕA (mSt ) = 0 as required. Proposition 3.12. Suppose that λ, µ, ν are partitions of n such that λ ν µ and that T T ΦT ∈ HomHT (M µ , S λ ) B B ΦB ∈ HomHB (M µ , S λ ) are such that T ˇ ΦT (mµT h) = 0 whenever h ∈ HT is such that mµT h ∈ Hµ and ˇ B ΦB (mµB h) = 0 whenever h ∈ HB is such that mµB h ∈ Hµ . Suppose that ΦT = ΓAT ϕAT AT ∈ T0 (λT , µT ) AT ΦB = ΓAB ϕAB AB ∈ T0 (λB , µB ) . AB Deﬁne the map Φ = (ΦT , ΦB ) as in Deﬁnition 3.8. Let S ∈ T0 (ν, µ), t ∈ Std(ν). Then Φ(mSt ) = 0. Proof. Assume that we have the conditions of Proposition 3.12. Using the notation of Lemma 3.10 ∗ mSt = Td(s) mν Td(t) s ∈ Std(ν) and µ(s) = S s = mµ hT hB Td(t) 10 ´ SINEAD LYLE AND ANDREW MATHAS for suitable hT ∈ HT , hB ∈ HB . Then Φ(mSt ) = ΓAT ΓAB ϕ(AT ,AB ) (mµ )hT hB Td(t) AT AB = Γ AT Γ AB ˇ Hλ + mλ Td(a) hT hB Td(t) a ∈ Std(λ) and µ(a) = (AT , AB ) AT AB a = Γ AT Γ AB ˇ Hλ + mλT Td(aT ) hT mλB Td(aB ) hB Td(t) AT AB aT aB aT ∈ Std(λT ) and µT (aT ) = AT , aB ∈ Std(λB ) and µB (aB ) = AB ˇ = Hλ + Γ AT mλT Td(aT ) hT Γ AB mλB Td(aB ) hB Td(t) . AT aT AB aB Note that ˇ T ΦT (mµT hT ) = Hλ + Γ AT mλT Td(aT ) hT AT aT B Φ (mµB h ) = H B ˇ λB + Γ AB mλB Td(aB ) hB . AB aB ˇ T ˇ B Now either mµT hT ∈ Hµ or mµB hB ∈ Hµ . Without loss of generality, suppose that m µT hT ∈ ˇ H µT . Then Γ AT ˇ T mλT Td(aT ) hT ∈ Hλ AT aT and B Γ AB mλB Td(aB ) hB ∈ M λ . AB aB Hence we can write ˇ Φ(mSt ) = Hλ + ∆xT yT mxT yT xT , y T ∈ Std(τ T ) where τ T λT xT ,y T ∆xB yB mxB yB xB , y B ∈ Std(τ B ) where τ B λB Td(t) xB ,y B for some ∆xT yT , ∆xB yB ∈ R. Now for any such mxT yT , mxB yB we may write ∗ ∗ mxT yT mxB yB = Td(xT ) mτ T Td(yT ) Td(xB ) mτ B Td(yB ) ∗ ∗ = Td(xT ) Td(xB ) mτ T mτ B Td(yT ) Td(yB ) ∗ = Td(xT ,xB ) mτ Td(yT ,yB ) T T B B where τ = (τ1 , τ2 , . . . , τ1 , τ2 , . . .) is a composition of n such that τ λ and (x T , xB ), (y T , y B ) are row standard τ -tableaux. By Deﬁnition 2.6, this can be written as an R-linear combination of terms muv where u, v ∈ Std(τ ) and τ ˇ ˇ τ λ. Hence mxT yT mxB yB ∈ Hλ . Since Hλ is a right ideal, it follows that mxT yT mxB yB Td(t) ∈ H ˇ λ and so Φ(m ) = 0 as required. St ROW AND COLUMN REMOVAL THEOREMS 11 T T B B Proposition 3.13. Suppose that ΦT ∈ HomHT (M µ , S λ ), and ΦB ∈ HomHB (M µ , S λ ) are such that ˇ T ΦT (mµT h) = 0 whenever h ∈ HT is such that mµT h ∈ Hµ and B ˇ ΦB (mµB h) = 0 whenever h ∈ HB is such that mµB h ∈ Hµ . ˆ Deﬁne the map Φ = (ΦT , ΦB ) as in Deﬁnition 3.8, and deﬁne the map Φ : S µ → S λ by ˆ ˇ Φ[(Hµ + mµ )h] = [Φ(mµ )]h for all h ∈ H. ˆ Then Φ is a well deﬁned H-homomorphism. ˇ Proof. It is suﬃcient to show that if h ∈ H is such that m µ h ∈ Hµ then [Φ(mµ )]h = Φ(mµ h) = 0. Suppose h satisﬁes these conditions. Then m µ h is an R-linear combination of terms of the form mSt where S ∈ T0 (ν, µ) and t ∈ Std(ν) for some ν µ. Choose such an m St . If λ ν then Φ(mSt ) = 0 by Proposition 3.11. If λ ν then Φ(m St ) = 0 by Proposition 3.12. Hence Φ(mµ h) = 0 as required. Deﬁnition 3.14. Suppose Φ ∈ Hom H (M µ , S λ ) is such that Φ(mµ h) = 0 for all h ∈ H such ˇ ˆ that mµ h ∈ Hµ . Deﬁne Φ ∈ HomH (S µ , S λ ) by setting ˆ ˇ Φ[(Hµ + mµ )h] = [Φ(mµ )]h for all h ∈ H. ˆ ˆ Similarly, if Φ ∈ HomH (S µ , S λ ), deﬁne Φ ∈ HomH (M µ , S λ ) by Φ = Φπ where π : M µ → S µ is the natural projection. We use analogous notation the for corresponding H T and HB homomorphisms. Deﬁnition 3.15. Suppose that T T ˆ ΦT ∈ HomHT (S µ , S λ ), B B ˆ ΦB ∈ HomHB (S µ , S λ ). Deﬁne T T ΦT ∈ HomHT (M µ , S λ ), B B ΦB ∈ HomHB (M µ , S λ ) as in Deﬁnition 3.14. Note that ΦT , ΦB satisfy the conditions of Proposition 3.13. Hence deﬁne ˆ ˆ ˆ Φ = (ΦT , ΦB ) ∈ HomH (S µ , S λ ) as in Proposition 3.13. We can now prove the main result of this section. Proof of Theorem 1.1. We have to show that T T B B HomH (S µ , S λ ) ∼ HomHT (S µ , S λ ) ⊗ HomHB (S µ , S λ ). = T T B B ˆ ˆ Let ΦT ∈ HomHT (S µ , S λ ) and ΦB ∈ HomHB (S µ , S λ ). Then there is an R-linear mapping T T B B from HomHT (S µ , S λ ) ⊗ HomHB (S µ , S λ ) into HomH (S µ , S λ ) given by sending ˆ ˆ ˆ ˆ ΦT ⊗ ΦB → ( ΦT , ΦB ) 12 ´ SINEAD LYLE AND ANDREW MATHAS and extending linearly. By construction this mapping is clearly injective. It remains only to show that it is surjective. ˆ Take Φ ∈ HomH (S µ , S λ ) and form the map Φ ∈ Hom H (M µ , S λ ). Then there exist ΓAT AB ∈ R such that Φ= ΓAT AB ϕ(AT ,AB ) . AT ,AB ˇ T ˇ Choose hT ∈ HT such that mµT hT ∈ Hµ . It therefore follows that mµ hT ∈ Hµ , and so, by construction of the map Φ, we have that Φ(m µ hT ) = 0. Thus Γ AT AB mλT Td(aT ) hT mλB Td(aB ) aT ∈ Std(λT ) and µT (aT ) = AT , AT ,AB aT ,aB aB ∈ Std(λB ) and µB (aB ) = AB ˇ ∈ Hλ . Hence for ﬁxed AB , it is necessary that Γ AT AB mλT Td(aT ) hT AT aT can be expressed in terms of muv where u, v ∈ Std(ν T ) and ν T λT . Hence Γ AT AB ˇ T mλT Td(aT ) hT ∈ Hλ . AT aT So for ﬁxed AB , the map ΦT = Γ AT AB ϕ AT AT ˆ T T ˆ is such that the corresponding map ΦT : S µ → S λ is well deﬁned. Suppose that {φi i ∈ I} T T form a basis of HomHT (S µ , S λ ). So we can write ˆ ΦT = ˆ βi,AB φi i and so Φ= βi,AB (φi , ϕAB ). i,AB A similar argument for ﬁxed φi shows that in fact Φ= αij (φT , φB ) i j i,j ˆ T T ˆ B B where φT ∈ HomHT (S µ , S λ ) and φB ∈ HomHB (S µ , S λ ) for all i, j; thus under our mapping, i j ˆ Φ appears as the image of T T B B ˆ ˆ αij (φT ⊗ φB ) ∈ HomHT (S λ , S µ ) ⊗ HomHB (S λ , S µ ). i j i,j ROW AND COLUMN REMOVAL THEOREMS 13 4. The q-Schur algebra Let R be a ring and q an invertible element of R; we allow q = −1. In this section we prove the following analogues of Theorems 1.1 and 1.2 for homomorphisms between Weyl modules. Theorem 4.1. Fix integers n, m with 0 ≤ m ≤ n and positive integers d, d , d . Let λ, µ ∈ Λ+ (d, n) and suppose that for some s we have λ1 + · · · + λs = µ1 + · · · + µs = n − m. Deﬁne µT , µB , λT , λB as in Theorem 1.1 and suppose that µT ∈ Λ+ (d , n − m), λT ∈ Λ+ (d , n − m), µB ∈ Λ+ (d , m), λB ∈ Λ+ (d , m). Then we have T T µB B HomS(d,n) (W µ , W λ ) ∼ HomS(d ,n−m) (W µ , W λ ) ⊗ HomS(d = ,m) (W , W λ ). Theorem 4.2. Fix integers n, l with 0 ≤ l ≤ n and positive integers d, d , d . Let λ, µ ∈ Λ+ (d, n) and suppose that for some r we have λ1 + · · · + λr = µ1 + · · · + µr = n − l. Deﬁne λ L , λR , µL , µR as in Theorem 1.2 and suppose that µL ∈ Λ+ (d , n − l), λL ∈ Λ+ (d , n − l), µR ∈ Λ+ (d , l), λR ∈ Λ+ (d , l). Then we have L L λR R HomS(d,n) (W λ , W µ ) ∼ HomS(d ,n−l) (W λ , W µ ) ⊗ HomS(d = ,l) (W , W µ ). Before we begin the proof, we note that in almost all cases these two results follow directly from Theorem 1.1 and Theorem 1.2 because Dipper and James [3, Cor. 8,6] have shown that HomH (S λ , S µ ) ∼ HomS(d,n) (W λ , W µ ) = whenever q = −1 or µ is 2-restricted. We give a direct argument so as to cover these cases as well. The next lemma shows that we can reduce to the case where d ≥ n, d ≥ n − l and d ≥ l. Lemma 4.3. Suppose that λ, µ ∈ Λ+ (d, n). Then HomS(d,n) (W λ , W µ ) ∼ HomS(d+1,n) (W λ , W µ ). = Proof. This follows from the general theory of Schur functors, as can be found in [1, §3.1]. To see this let ϕd = {ϕα | α ∈ Λ(d, n)} ∈ S(d + 1, n). Then ϕd is an idempotent so P = ϕd S(d + 1, n) is a projective S(d + 1, n)-module. Since S(d, n) ∼ ϕd S(d + 1, n)ϕd we can deﬁne a functor = α : mod- S(d + 1, n) −→ mod- S(d, n) by α = Hom S(d+1,n) (P, ). If λ, µ ∈ Λ+ (d, n) then every composition factor of the S(d + 1, n)-modules W λ and W µ are contained in the head of P , so Hom S(d+1,n) W λ , W µ ∼ HomS(d,n) α(W λ ), α(W µ ) by [1, = Cor. 3.1c]. It is easy to see that if ν ∈ Λ(d, n) then α(W ν ) ∼ W ν , as S(d, n)-modules, so this = completes the proof. To show that Theorem 4.1 and Theorem 4.2 are equivalent we need some more notation. Fix an integer d ≥ n and for λ ∈ Λ(d, n) deﬁne (w) nλ = (−q)− Tw . w∈Sλ 14 ´ SINEAD LYLE AND ANDREW MATHAS Set N λ = nλ H and let SR = SR (d, n) be the endomorphism algebra SR = EndH M µ . µ∈Λ(d,n) Now N λ = (M λ )# , where # is the automorphism used in the proof of Lemma 3.1. Therefore, HomH (M µ , M λ ) ∼ HomH (N µ , N λ ) and, consequently, # induces a canonical algebra isomor- = # phism β : S(d, n) − SR (d, n); explicitly, β φ (h) = φ(h# ) , for all φ ∈ S(d, n) and all → h ∈ H. For λ ∈ Λ+ (d, n) deﬁne WR = β(W λ ). Then WR is an analogue of a Weyl module for λ λ the algebra SR (d, n). Donkin has shown that if d ≥ n then SR (d, n) is the Ringel dual of the q-Schur algebra S(d, n) [5]. Consequently, we have a contravariant functor γ which sends S(d, n)-modules to SR (d, n)-modules. Further, if ν ∈ Λ(d, n) then γ(W ν ) ∼ (WR ) , where (WR ) is the contragre- = ν ν dient dual of WRν . Proofs and further explanations of these results can be found in [5]. Lemma 4.4. Suppose that d ≥ n and λ, µ ∈ Λ + (d, n) Then HomS(d,n) (W µ , W λ ) ∼ HomS(d,n) (W λ , W µ ). = Proof. Since d ≥ n, SR (d, n) is the Ringel dual of S(d, n). Therefore, using the functors above, we have HomS(d,n) (W µ , W λ ) ∼ HomSR (d,n) WR , WR , = µ λ by applying β, ∼ HomS (d,n) (W λ ) , (W ) , = µ by duality, R R R ∼ HomS(d,n) W , W µ , = λ where the last line follows by applying γ. Proposition 4.5. Theorem 4.1 holds if and only if Theorem 4.2 holds. Proof. By Lemma 4.3 we may assume that d, d , d are all large enough so that we can apply Lemma 4.4. The result now follows by the argument of Proposition 3.2 We now prove Theorem 4.1. The proof is closely related to the proof of Theorem 1.1. Fix integers n, m with 0 ≤ m ≤ n, and positive integers d, d , d . Fix λ, µ ∈ Λ+ (d, n) where λ1 + . . . + λ s = µ 1 + . . . + µ s = n − m for some s and λT , µT ∈ Λ+ (d , n − m) λB , µB ∈ Λ+ (d , m) We use the deﬁnitions of H T ∼ Hn−m and HB ∼ Hm from Deﬁnition 3.3 and deﬁne = = S = EndH Mτ τ ∈Λ(d,n) to be an endomorphism algebra of H, S T = EndHT Mτ τ ∈Λ(d ,n−m) ROW AND COLUMN REMOVAL THEOREMS 15 to be an endomorphism algebra of H T and S B = EndHB Mτ τ ∈Λ(d ,m) T T to be an endomorphism algebra of H B . Then modules such as Mτ , W τ are deﬁned in the obvious way. Deﬁnition 4.6. Let T T θT = ΓAT θAT AT ∈ T0 (λT , µT ) ∈ HomS T (Mµ , W λ ) AT B B θB = ΓAB θAB AB ∈ T0 (λB , µB ) ∈ HomS B (Mµ , W λ ) AB and deﬁne θ = (θ T , θ B ) : Mµ → W λ by θ= ΓAT ΓAB θ(AT ,AB ) . AT AB Note that by Lemma 2.17, this deﬁnes a bijection T T B B HomS T (Mµ , W λ ) ⊗ HomS B (Mµ , W λ ) → HomS (Mµ , W λ ). Proposition 4.7. Let ν ∈ Λ+ (d, n), τ ∈ Λ(d, n) be such that λ ν µ. Let U ∈ T 0 (ν, µ), V ∈ T0 (ν, τ ), A ∈ T0 (λ, µ). Then θA (ϕU V ) = 0. Proof. Deﬁne θT λ A ∈ HomS (Mµ , Mλ ) by θT λ A (ϕµ f ) = ϕT λ A f , for all f ∈ S. Then θT λ A is a well deﬁned S-homomorphism and θA = πθT λ A where π : Mλ → W λ is the natural projection. Note that ϕU V = ϕµ ϕU V . Then θT λ A (ϕU V ) = (θT λ A (ϕµ ))ϕU V = ϕ T λ A ϕU V . Now ϕT λ A ∈ S λ and ϕU V ∈ S ν . So, since λ ˇ ν, we have that ϕT λ A ϕU V ∈ S λ . Hence θA (ϕU V ) = π(ϕT λ A ϕU V ) = 0. Proposition 4.8. Suppose that ν ∈ Λ+ (d, n) is such that λ ν µ and τ ∈ Λ(d, n). Let U ∈ T0 (ν, µ), V ∈ T0 (ν, τ ). Suppose that T T θ T ∈ HomS T (Mµ , W λ ) B B θ B ∈ HomS B (Mµ , W λ ) are such that T ˇ θ T (ϕµT f ) = 0 whenever f ∈ S T is such that ϕµT f ∈ S µ and B ˇ θ B (ϕµB f ) = 0 whenever f ∈ S B is such that ϕµB f ∈ S µ . 16 ´ SINEAD LYLE AND ANDREW MATHAS Suppose that θT = ΓAT θAT AT ∈ T0 (λT , µT ) AT θB = ΓAB θAB AB ∈ T0 (λB , µB ) . AB Deﬁne the map θ = (θ T , θ B ) as in Deﬁnition 4.6. Then θ(ϕU V ) = 0. Proof. Note that ϕU V = ϕµ ϕU V . Then θ(ϕU V ) = ΓAT ΓAB θ(AT ,AB ) (ϕµ )ϕU V AT AB = ΓAT ΓAB ϕ(AT ,AB ) ϕU V ∈ HomH (M τ , S λ ). AT AB Then, using the notation of Lemma 3.10, there exist h T ∈ HT , hB ∈ HB such that [θ(ϕU V )](mτ ) = ΓAT ΓAB ϕ(AT ,AB ) (ϕU V (mτ )) AT AB = ΓAT ΓAB ϕ(AT ,AB ) AT AB ∗ Td(u) mν Td(v) u, v ∈ Std(ν), µ(u) = U, τ (v) = V u v = ΓAT ΓAB ϕ(AT ,AB ) mµ hT hB Td(v) AT AB v ˇ = Hλ + ΓAT ΓAB mλT mλB Td(aT ) Td(aB ) hT hB Td(v) AT AB aT aB v aT ∈ Std(λT ), aB ∈ Std(λB ), µT (aT ) = AT , µB (aB ) = AT ˇ T ˇ B where either ν T µT and mµT hT ∈ Hµ or ν B µB and mµB hB ∈ Hµ . Without loss of generality, ˇ T T T assume mµT hT ∈ Hµ . Consider ϕU T T ν T ∈ HomH (M ν , M µ ). Then θ T (ϕU T T ν T ) = 0 by assumption and hence T ˇ ΓAT mλT Td(aT ) hT = [θ T (ϕAT T ν T )](mν T ) ∈ Hλ . A T aT Also, B ΓAT mλB Td(aB ) hB ∈ M λ . A T aB Proposition 4.8 now follows from the proof of Proposition 3.13. T T B B Proposition 4.9. Suppose that θ T ∈ HomS T (Mµ , W λ ), and θ B ∈ HomS B (Mµ , W λ ) are such that T ˇ θ T (ϕµT f ) = 0 whenever f ∈ S T is such that ϕµT f ∈ S µ ROW AND COLUMN REMOVAL THEOREMS 17 and ˇ B θ B (ϕµB f ) = 0 whenever f ∈ S B is such that ϕµB f ∈ S µ . ˆ Deﬁne the map θ = (θ T , θ B ) as in Deﬁnition 4.6, and deﬁne the map θ : W µ → W λ by ˆ ˇ θ[(S µ + ϕµ )f ] = [θ(ϕµ )]f for all f ∈ S. ˆ Then θ is a well deﬁned S-homomorphism. ˇ Proof. It is suﬃcient to show that if f ∈ S is such that ϕ µ f ∈ S µ then [θ(ϕµ )]f = θ(ϕµ f ) = 0. Suppose f satisﬁes these conditions. Then ϕ µ f is an R-linear combination of terms of the form ϕU V where U ∈ T0 (ν, µ) and V ∈ T0 (ν, τ ) for some ν ∈ Λ+ (d, n), τ ∈ Λ(d, n) such that ν µ. Choose such an ϕU V . If λ ν then θ(ϕU V ) = 0 by Proposition 4.7. If λ ν then θ(ϕ U V ) = 0 by Proposition 4.8. Hence θ(ϕµ f ) = 0 as required. Deﬁnition 4.10. Suppose that T T ˆ θ T ∈ HomS T (W µ , W λ ), B B ˆ θ B ∈ HomS B (W µ , W λ ). Deﬁne T T θ T ∈ HomS T (Mµ , W λ ), B B θ B ∈ HomS B (Mµ , W λ ). ˆ ˆ in the obvious manner; that is, θ T = θ T π T and θ B = θ B π B where πT and πB are the projections onto W µT and W µB respectively. Note that θ T and θ B satisfy the conditions of Proposition 4.9. Hence deﬁne ˆ ˆ ˆ θ = (θ T , θ B ) ∈ HomS (W µ , W λ ) as in Proposition 4.9. Proposition 4.11. Let θ ∈ HomS (Mµ , W λ ) be such that ˇ θ(ϕµ f ) = 0 for any f ∈ S such that ϕµ f ∈ S µ . Then there exist ΓAT AB ∈ R such that θ= ΓAT AB θ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB ) . AT ,AB Fix AB and consider T T θT = ΓAT AB θAT ∈ HomS T (Mµ , W λ ). AT Then ˇ T θ T (ϕµT f ) = 0 for any f ∈ S T such that ϕµT f ∈ S µ . Proof. Suppose we have the conditions of Proposition 4.11. By Proposition 4.7 and Proposition T 4.9, it is suﬃcient to show that θ T (ϕU V ) = 0 for any U ∈ T0 (ν T , µT ), V = T ν such that ν T ∈ Λ+ (d, n) and λT ν T µT . We have Td(u) mν T u ∈ Std(ν T ) and µT (u) = U ∗ = m µT h T u 18 ´ SINEAD LYLE AND ANDREW MATHAS for some hT ∈ HT . Then θ T (ϕU V ) = 0 if and only if Γ AT AB mλT Td(aT ) hT aT ∈ Std(λT ) and µT (at ) = AT ˇ T ∈ Hλ . AT aT B Now ﬁx ν = (ν T , ν B ), X = (U T , T ν ), Y = T ν . By assumption, θ(ϕXY ) = 0. Hence Γ AT AB mλT mλB Td(aT ) Td(aB ) hT aB ∈ Std(λB ) and µB (aB ) = AB ˇ ∈ Hλ AT ,AB aT aB and so Γ AT AB ˇ T mλT Td(aT ) hT ∈ Hλ AT aT as required. Deﬁnition 4.12. Suppose θ ∈ HomH (Mµ , W λ ) is such that θ(ϕµ f ) = 0 for all f ∈ S such that ˇ ˆ ϕµ f ∈ S µ . Deﬁne θ ∈ HomS (W µ , W λ ) by setting ˆ ˇ θ[(S µ + ϕµ )f ] = [θ(ϕµ )]f for all f ∈ S. ˆ ˆ Similarly, if θ ∈ HomS (W µ , W λ ), deﬁne θ ∈ HomS (Mµ , W λ ) by θ = θπ where π : Mµ → W µ is the natural projection. We use analogous notation for the corresponding S T and S B homomorphisms. We can now prove our second main result. Proof of Theorem 4.1. We have to show that T T B B HomS (W µ , W λ ) ∼ HomS T (W µ , W λ ) ⊗ HomS B (W µ , W λ ). = ˆ T T ˆ B B Let θ T ∈ HomHT (W µ , W λ ) and θ B ∈ HomS B (W µ , W λ ). Then there is an R-linear mapping T T B B from HomS T (W µ , W λ ) ⊗ HomS B (W µ , W λ ) into HomS (W µ , W λ ) given by sending ˆ ˆ ˆ ˆ θ T ⊗ θ B → (θ T , θ B ) and extending linearly. By construction, this mapping is injective. It remains only to show that it is surjective. ˆ ˆ Let θ ∈ HomS (W µ , W λ ). Form θ = θπ ∈ HomS (Mµ , W λ ) so that θ satisﬁes the conditions of Proposition 4.11. Suppose θ= ΓAT AB θ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB ) AT ,AB for some ΓAT AB ∈ R. Fix AB . By Proposition 4.11, the map θT = Γ AT AB θ AT AT ˆ T T ˆ is such that the corresponding map θ T : W µ → W λ is well deﬁned. Suppose that {ψi | i ∈ I} is a basis of HomS T (W µT , W λT ). So we can write ˆ θT = ˆ βi,AB ψi i and so θ= βi,AB (ψi , θAB ). i,AB ROW AND COLUMN REMOVAL THEOREMS 19 A similar argument for ﬁxed ψi shows that in fact T B θ= αij (ψi , ψj ) i,j ˆT where ψi ∈ HomS T (W µT ,W λT ˆB B B ˆ ) and ψj ∈ HomS B (W µ , W λ ); thus under our mapping, θ appears as the image of T T B B ˆT ˆB αij (ψi ⊗ ψj ) ∈ HomS T (W λ , W µ ) ⊗ HomS B (W λ , W µ ). i,j References [1] J. Brundan, R. Dipper, and A. Kleshchev, Quantum linear groups and representations of GL n (Fq ), Memoirs A.M.S., 706, A.M.S., 2001. [2] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. L.M.S. (3), 52 (1986), 20–52. [3] , q–Tensor space and q–Weyl modules, Trans. A.M.S., 327 (1991), 251–282. [4] S. Donkin, A note of decomposition numbers for general linear groups and symmetric groups, Math. Proc. Cambridge Phil. Soc., 97 (1985), 57–62. [5] , On tilting modules for algebraic groups, Math. Z., 212 (1993), 39–60. [6] M. Fayers and S. Lyle, Row and column removal theorems for homomorphisms between Specht modules, J. Pure Appl. Algebra, 185 (2003), 147–164. [7] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1–34. [8] G. D. James, The decomposition matrices of GLn (q) for n ≤ 10, Proc. L.M.S. (3), 60 (1990), 225–264. [9] A. Mathas, Hecke algebras and Schur algebras of the symmetric group, Univ. Lecture Notes, 15, A.M.S., 1999. [10] G. E. Murphy, The representations of Hecke algebras of type An , J. Algebra, 173 (1995), 97–121. School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

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