Double coset algebras

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Double coset algebras Robert May Department of Mathematics and Computer Science, Longwood University, 201 High Street, Farmville, VA 23909, United States

a r t i c l e

i n f o

Article history: Received 22 July 2013 Received in revised form 17 February 2014 Available online xxxx Communicated by C. Kassel MSC: 16S99; 20C08; 20M25

a b s t r a c t For a finite monoid M with unit e and an indexed family G = {Gi : i ∈ I} of subgroups of the group G(M ) of invertible elements in M , the complex vector space A(M, G) with basis the double cosets of the form Gi mGj , m ∈ M , has a natural multiplication yielding an associative C-algebra which we call a double coset algebra. We construct two Z-algebras with identity, LGS(M, G) and RGS(M, G), called the left and right generalized Schur algebras, which as Z-modules are free with basis the double cosets. When M is the symmetric group Sr and G is the family of Young subgroups indexed by compositions of r with at most n parts, A(M, G) is isomorphic to the usual Schur algebra S(n, r) and LGS(M, G) corresponds to its standard Z-form. The structure constants we derive for these algebras provide multiplication rules useful for further analysis of the generalized Schur algebras. Our main result is then the observation that LGS(M, G) and RGS(M, G) are always Z-forms for A(M, G), that is, A(M, G) ∼ = C ⊗Z RGS(M, G). = C ⊗Z LGS(M, G) ∼ The Iwahori–Hecke algebra H(M, G) corresponding to a finite monoid M and subgroup G is isomorphic to the double coset algebra A(M, G) where G = {G} consists of a single set. So, as a special case of our result, we obtain Z-forms LGS(M, G) and RGS(M, G) for arbitrary Iwahori–Hecke algebras. The existence of a Z-form for any such H(M, G) appears to be a new result. © 2014 Elsevier B.V. All rights reserved.

1. Introduction For a finite monoid M with unit e and an indexed family G = {Gi : i ∈ I} of subgroups of the group G(M ) of invertible elements in M consider the complex vector space A(M, G) with basis the double cosets of the form Gi mGj , m ∈ M . There is a natural multiplication on A(M, G) yielding an associative C-algebra which we call a double coset algebra. We also construct two Z-algebras with identity, LGS(M, G) and RGS(M, G), which we call left and right generalized Schur algebras. As Z-modules, these are free with basis the double cosets. When M is the symmetric group Sr or the full transformation semigroup τr and G is the family of Young subgroups indexed by compositions of r with at most n parts, A(M, G) is isomorphic to the E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2014.03.006 0022-4049/© 2014 Elsevier B.V. All rights reserved.

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usual Schur algebra S(n, r) or the generalized Schur algebra of [5] and [4]. Then LGS(M, G) corresponds to the known Z-form for the usual Schur algebra, while RGS(M, G) corresponds to the Z-form in [4] for a generalized Schur algebra. We derive structure constants for these algebras, providing useful multiplication rules for the Schur algebras. Our main result is then the observation that LGS(M, G) and RGS(M, G) are always Z-forms for A(M, G), that is, A(M, G) ∼ = C ⊗Z RGS(M, G). = C ⊗Z LGS(M, G) ∼ The Iwahori–Hecke algebra H(M, G) corresponding to a finite monoid M and subgroup G (see [2,6,7]) is isomorphic to the double coset algebra A(M, G) where G = {G} consists of a single set. So, as a special case of our result, we obtain Z-forms LGS(M, G) and RGS(M, G) for arbitrary Iwahori–Hecke algebras. When M is itself a group the two Z-forms are isomorphic and agree with the well-known Z-form for this case. For general finite monoids M , the two Z-forms are in general distinct and the existence of a Z-form for any such H(M, G) appears to be a new result. We remark that the Z-forms obtained by Solomon in [7] and Godelle in [2] for special cases when A(M, G) is the q-rook algebra or a deformation of the monoid algebra corresponding to a Renner monoid differ from either of our Z-forms. 2. The standard double coset algebra Let M be a finite monoid with identity e and G = {Gi : i ∈ I} be a family of subgroups of the group G(M ) of invertible elements in M indexed by a finite set I. (We may have Gi = Gj for distinct indices i = j.) For each i, j ∈ I there is an equivalence relation i ∼j on M , defined by mi ∼j n ⇔ n = σmπ for some σ ∈ Gi , π ∈ Gj , for which the equivalence classes are the double cosets of the form Gi mGj , m ∈ M . Let i Mj be the set of such double cosets. Let MI be the disjoint union of all the sets i Mj , i, j ∈ I. More precisely, MI = {(D, i, j): i, j ∈ I, D ∈ i Mj }. Let A = C[M ] be the monoid algebra over the complex numbers C and let i Aj be the subspace of A which is invariant under left multiplication by Gi and right multiplication by Gj : i

Aj = {x ∈ A: ∀σ ∈ Gi , ∀π ∈ Gj , σxπ = x}.

 For any double coset D ∈ i Mj define X(D) = m∈D m ∈ A. Evidently X(D) ∈ i Aj and a simple calculation shows that {X(D): D ∈ i Mj } is a basis for the subspace i Aj ⊆ C[M ]. (For example, see [2] for the special case of Iwahori–Hecke algebras.) Notice that the subspaces i Aj for distinct indices may have nontrivial intersection or may even coincide. We take disjoint copies of these subspaces and form the direct sum: A(M, G) ≡ AI ≡



i

Aj .

i,j∈I

Then the vector space AI has a basis {X(D): (D, i, j) ∈ MI } indexed by MI . More generally, let r : MI → C∗ be a scaling function which assigns to each double coset (D, i, j) ∈ MI a nonzero value r(D, i, j) ∈ C−{0}. Then the set {r(D, i, j)X(D): (D, i, j) ∈ MI } forms a basis for AI . We will write br (D, i, j) = r(D, i, j)X(D) for these basis elements. We wish to make AI into an associative algebra with identity. The construction is a natural generalization of the special case of Iwahori–Hecke algebras as given in [2]. For i ∈ I we have Gi = Gi eGi ∈ i Mi . Then i) i i define ei = X(G o(Gi ) ∈ A ⊆ A, where o(Gi ) is the order of the group Gi . It is easy to check that each ei is an idempotent in A and that i Aj = ei Aej . We want the idempotents ei to be orthogonal in AI which is equivalent to taking x ∗ y = 0 when x ∈ i Aj , y ∈ k Al and j = k. For the case j = k, notice that

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x ∈ i Aj ⊆ A,

y ∈ k Al ⊆ A

⇒

3

xy ∈ i Al

(where the product xy is the product in A). So a product on AI is given by  x∈ A , i

j

y∈ A k

l

⇒

x∗y =

0 xy ∈ i Al

if j = k if j = k.

This product is easily seen to be associative and bilinear, since the multiplication in the algebra A is. Since  the idempotents ei are orthogonal, it is easy to check that i∈I ei is an identity for the algebra AI . Definition 2.1. The standard (M, G) double coset algebra (DCA) (over C) is the algebra AI with the product given by  x∈ A , i

j

y∈ A k

l

⇒

x∗y =

0 xy ∈ i Al

if j = k if j = k.

We have seen the following Proposition 2.1. AI is an associative C-algebra with identity 1 = MI } is a basis for AI for any scaling function r.

 i∈I

ei . The set {br (D, i, j): (D, i, j) ∈

3. Structure constants for AI We now describe structure constants for the standard DCA AI relative to a basis {br (D, i, j): (D, i, j) ∈ MI }. We have br (D1 , i, j)br (D2 , k, l) =



ar (D1 , i, j, D2 , k, l, D)br (D, i, l)

(D,i,l)∈MI

for certain ar (D1 , i, j, D2 , k, l, D) ∈ C. ar (D1 , i, j, D2 , k, l, D) = 0 when j = k by orthogonality. It is not hard to check the following lemma (which generalizes the Iwahori–Hecke algebra case as given in [2]): Lemma 3.1. Let D ∈ i Mk and take any m ∈ D. Then ar (D1 , i, j, D2 , j, k, D) =

r(D1 , i, j)r(D2 , j, k) a(D1 , i, j, D2 , j, k, D) r(D, i, k)

where a(D1 , i, j, D2 , j, k, D) is a nonnegative integer given by   a(D1 , i, j, D2 , j, k, D) = # (m1 , m2 ) ∈ D1 × D2 : m1 m2 = m . Notice that a(D1 , i, j, D2 , j, k, D) = a1 (D1 , i, j, D2 , j, k, D) where the scaling function 1 takes each double coset to 1: 1(D, i, j) = 1 for any (D, i, j) ∈ MI . There is an alternative formula for these integers a. For any double coset D ∈ i Mj , the product group Gi × Gj acts transitively on D by (σ, π)(m) = σmπ −1 for m ∈ D, σ ∈ Gi , π ∈ Gj . Then for any m ∈ D, let n(D) be the order of the subgroup Gi,j,m of Gi × Gj which leaves m fixed:   n(D) = # (σ, π): σ ∈ Gi , π ∈ Gj , σmπ −1 = m .

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Gi,j,m , Gi,j,n are conjugate subgroups whenever m, n ∈ D, so the order n(D) is independent of the choice of m ∈ D. The cosets (α, β)Gi,j,m of Gi,j,m in Gi × Gj correspond one to one with elements αmβ −1 ∈ D = Gi mGj  and each such coset contains exactly n(D) elements. For i ∈ I, recall X(Gi ) = X(Gi eGi ) = σ∈Gi σ =  −1 . Then we have σ∈Gi σ Lemma 3.2. X(Gi )mX(Gj ) = n(D)X(D) where D = Gi mGj . Next, for D1 ∈ i Mj , D2 ∈ j Mk , D ∈ i Mk and m ∈ D1 , n ∈ D2 , put N (D1 , D2 , D) = #{ρ ∈ Gj : mρn ∈ D} (which is independent of the choices of m, n). Finally, let o(Gj ) be the order of the group Gj . Then Lemma 3.3. For the standard DCA AI , the nonnegative integers a are given by a(D1 , i, j, D2 , j, k, D) =

o(Gj )N (D1 , D2 , D)n(D) . n(D1 )n(D2 )

Proof. Using Lemma 3.2, compute n(D1 )n(D2 )X(D1 )X(D2 ) = X(Gi )mX(Gj )X(Gj )nX(Gk ) = o(Gj )



X(Gi )mρnX(Gk )

ρ∈Gj

Let D(mρn) = Gi mρnGk ∈ i Mk . Then by Lemma 3.2, X(Gi )mρnX(Gk ) = n(D(mρn))X(D(mρn)). For each D ∈ i Mk there are exactly N (D1 , D2 , D) values of ρ ∈ Gj such that D(mρn) = D. Then o(Gj )

 ρ∈Gj

= o(Gj )

X(Gi )mρnX(Gk ) 

   n D(mρn) X D(mρn) = o(Gj )N (D1 , D2 , D)n(D)X(D).

ρ∈Gj

D∈i Mk

But (recalling that X(D1 ) = b1 (D1 , i, j), a = a1 , etc.) we also have n(D1 )n(D2 )X(D1 )X(D2 ) =



n(D1 )n(D2 )a(D1 , i, j, D2 , j, k, D)X(D).

D∈i Mk

Since the X(D) are independent, we can equate the integer coefficients and find that n(D1 )n(D2 ) must divide each coefficient o(Gj )N (D1 , D2 , D)n(D). Then dividing by n(D1 )n(D2 ) gives a(D1 , i, j, D2 , j, k, D) = o(Gj )N (D1 ,D2 ,D)n(D) as required. 2 n(D1 )n(D2 ) 4. Z -forms for AI If the structure constants ar (D1 , i, j, D2 , j, k, D) for AI relative to a scaling function r are all integers, we can define an associative Z-algebra AZI,r as follows: AZI,r is a free Z-module with a basis {bD,i,j : (D, i, j) ∈ MI } indexed by MI . A bilinear product on AZI,r is determined by bD1 ,i,j ∗ bD2 ,j,k =

 D∈i Mk

ar (D1 , i, j, D2 , j, k, D)bD,i,k

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where the structure constants ar (D1 , i, j, D2 , j, k, D) are assumed to be integers. The associative law will hold for this product if and only if (bD1 ∗ bD2 ) ∗ bD3 = bD1 ∗ (bD2 ∗ bD3 ) for all D1 , D2 , D3 ∈ MI . A little computation shows that this is equivalent to the following associative condition: 

ar (D1 , D2 , D)ar (D, D3 , D4 ) =

D∈MI



ar (D2 , D3 , D)ar (D1 , D, D4 )

D∈MI

for all D1 , D2 , D3 , D4 ∈ MI . But the structure constants ar come from the known associative algebra AI , so this associative condition must be satisfied and the product on AZI,r must be associative as well. The algebra AZI,r will in general not have a unit. For example, the structure constants a1 = a for AI with the scaling function r = 1 are all integers by Lemma 3.1, but the corresponding AZI,1 in general does not contain an identity. (An idempotent ei is not in AZI,1 except in the trivial case when o(Gi ) = 1.)  Evidently AZI,r will contain an identity if and only if the identity i∈I ei = 1 in AI can be written as a linear combination of the basis elements br (D, i, j) with integer coefficients. Definition 4.1. If the structure constants ar for AI relative to a scaling function r and corresponding basis elements br (D, i, j) are all integers and if the identity in AI is a linear combination of the basis elements br (D, i, j) with integer coefficients, then the corresponding Z -algebra with unit AZI,r is a Z-form for AI . Notice that for any Z-form AZI,r , the correspondence br (D, i, j) ↔ 1 ⊗ bD,i,j between basis elements gives an isomorphism AI ∼ = C ⊗Z AZI,r since the structure constants are identical. In this section we will describe Z-forms for any AI . Let i M be the set of left Gi cosets in M of the form Gi m, m ∈ M . Similarly, let Mj be the set of right cosets of Gj in M . Any double coset D ∈ i Mj is a disjoint union of left cosets C ∈ i M and any two such left cosets contain the same number of elements. Define nL (D) to be the number of elements in any left coset C ∈ i M , C ⊆ D. Similarly, define nR (D) to be the number of elements in any right coset C ∈ Mj , C ⊆ D. Next note that Gj acts transitively on the set of left cosets C ∈ i M , C ⊆ D. Let Gj,C = {σ ∈ Gj : Cσ = C} be the subgroup which fixes a given C. Define nL (D, j) to be the order of Gj,C . For any two left cosets C1 , C2 ∈ i M , C1 , C2 ⊆ D the subgroups Gj,C1 , Gj,C2 are conjugate, so nL (D, j) is independent of the choice of C. Similarly, let GC,i = {σ ∈ Gi : σC = C} be the subgroup of Gi which fixes a given right coset C ∈ Mj , C ⊆ D and let nR (D, i) be the order of GC,i (independent of the choice of C). Lemma 4.1. nL (D, j) =

nL (D)n(D) o(Gi )

for any D ∈ i Mj .

Proof. We count #D, the number of elements in the double coset D, in two ways. First, since Gi × Gj acts transitively on D with n(D) the order of the subgroup leaving an element of D fixed, we have #D = o(Gi ×Gj ) o(Gi )o(Gj ) = . On the other hand, #D equals the number of left cosets contained in D times the n(D) n(D) number of elements, nL (D), in each left coset. Since Gj acts transitively on the left Gi cosets contained in D and nL (D, j) is the order of the subgroup leaving a left coset fixed, the total number of left cosets o(Gj ) o(Gj ) . Then #D = nL (D,j) nL (D). Equating the two expressions for #D and contained in D is given by nL (D,j) solving for nL (D, j) gives the result. 2 Lemma 4.2. For the scaling factor L(D, i, j) = ture constants for AI are given by

1 nL (D)

aL (D1 , i, j, D2 , j, k, D) =

and corresponding basis vectors bL (D, i, j), the struc-

nL (D, k)N (D1 , D2 , D) . nL (D1 , j)nL (D2 , k)

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Proof. By Lemmas 3.1 and 3.3, aL (D1 , i, j, D2 , j, k, D) =

nL (D) o(Gj )N (D1 , D2 , D)n(D) nL (D) a(D1 , i, j, D2 , j, k, D) = . nL (D1 )nL (D2 ) nL (D1 )nL (D2 ) n(D1 )n(D2 )

Then using Lemma 4.1 gives aL (D1 , i, j, D2 , j, k, D) =

N (D1 , D2 , D)nL (D, k) o(Gj )N (D1 , D2 , D)nL (D, k)o(Gi ) = . nL (D1 , j)o(Gi )nL (D2 , k)o(Gj ) nL (D1 , j)nL (D2 , k)

2

Lemma 4.3. For any D1 ∈ i Mj , D2 ∈ j Mk , D ∈ i Mk , i, j, k ∈ I, the constant aL (D1 , i, j, D2 , j, k, D) = nL (D,k)N (D1 ,D2 ,D) nL (D1 ,j)nL (D2 ,k) is an integer. We defer the proof of this key lemma to the next section. Theorem 4.1. The Z-algebra AZI,L corresponding to the scaling function L and basis {bL (D, i, j): (D, i, j) ∈ MI } for AI is a Z-form for AI with structure constants aL (D1 , i, j, D2 , j, k, D) =

nL (D,k)N (D1 ,D2 ,D) nL (D1 ,j)nL (D2 ,k) .

Proof. By Lemma 4.3, the structure constants are all integers, so to verify that this AZI,L is in fact a Z-form we only need to show that the identity in AI is an integral combination of the basis elements. The identity  i) in AI is i∈I ei where ei = X(G o(Gi ) . But the only left Gi coset in the double coset Gi = Gi eGi is Gi itself, X(Gi ) i) which has o(Gi ) elements. So nL (Gi ) = o(Gi ). Then ei = X(G o(Gi ) = nL (Gi ) = L(Gi )X(Gi ) = bL (Gi , i, i),   a basis element. So the identity 1 = i∈I ei = i∈I bL (Gi , i, i) is indeed an integral combination of basis elements. 2

We call AZI,L the left standard Z-form for AI . We can obtain a second Z-form with unit for AI by replacing nL by nR in the above results. The proofs of these “right” results are exactly parallel to the proofs of the “left” results and are omitted. Lemma 4.4. nR (D, i) =

nR (D)n(D) o(Gj )

for any D ∈ i Mj .

Lemma 4.5. For the scaling factor R(D, i, j) = ture constants for AI are given by

1 nR (D)

aR (D1 , i, j, D2 , j, k, D) =

and corresponding basis vectors bR (D, i, j), the struc-

nR (D, k)N (D1 , D2 , D) . nR (D1 , j)nR (D2 , k)

Lemma 4.6. For any D1 ∈ i Mj , D2 ∈ j Mk , D ∈ i Mk , i, j, k ∈ I, the constant aR (D1 , i, j, D2 , j, k, D) = nR (D,k)N (D1 ,D2 ,D) nR (D1 ,j)nR (D2 ,k) is an integer. Theorem 4.2. The Z-algebra AZI,R corresponding to the scaling function R and basis {bR (D, i, j): (D, i, j) ∈ MI } for AI is a Z-form for AI with structure constants aR (D1 , i, j, D2 , j, k, D) =

nR (D,k)N (D1 ,D2 ,D) nR (D1 ,j)nR (D2 ,k) .

We call AZI,R the right standard Z-form for AI . 5. Generalized Schur algebras and construction of Z -forms In this section we will prove Lemmas 4.3 and 4.6 (completing the proofs of Theorems 4.1 and 4.2). To show that the structure constants aL and aR are integers, we first define certain endomorphism algebras,

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the left and right “generalized Schur algebras”, which will be Z-algebras with unit. We show that these algebras are free as Z-modules with bases corresponding to the double cosets MI . We then show that the structure constants for these algebras (which are necessarily in Z) are just the aL (D1 , i, j, D2 , j, k, D) = nL (D,k)N (D1 ,D2 ,D) nR (D,k)N (D1 ,D2 ,D) nL (D1 ,j)nL (D2 ,k) and aR (D1 , i, j, D2 , j, k, D) = nR (D1 ,j)nR (D2 ,k) of the left and right standard Z-forms. i First, let A = Z[M ] and let A be the free Z-module with basis i M , the set of left Gi -cosets in M . For any left coset C ∈ i M and any m ∈ M , Cm is another left coset. Evidently (Cm1 )m2 = C(m1 m2 ), so Ai can be considered as a right A-module. Notice that Ai is a principal A-module, Ai = Gi eA for the left coset Gi e = G i ∈ i M .

i I I Now define AI = i∈I A . Then A is a right A-module and a free Z-module with basis b(A ) ≡ I I I {(C, i): i ∈ I, C ∈ i M }. Let End Z (A ) be the set of Z-linear endomorphisms of A . End Z (A ) is a Z-algebra with product the composition of functions and unit the identity map. There is an injective Z-linear map ψ : A → End Z (AI ) given by ψ(a) = fa where fa (x) = xa for x ∈ AI , a ∈ A. ψ is an algebra anti-homomorphism, so we can regard the opposite algebra Aop as a subalgebra of End Z (AI ) and will define B ≡ BL as the “commuting algebra” of Aop : Definition 5.1. The left generalized Schur algebra, LGS Z (M, G), is the Z-algebra B ≡ BL ≡ End Aop (AI ). For any domain R, the left generalized Schur algebra over R is the R-algebra LGS R (M, G) = R⊗Z LGS Z (M, G). Note: We will justify this definition in Section 7 by showing that the usual Schur algebra Sk (r, n) is isomorphic to LGS k (M, G) for M = Sr and G the set of all Young subgroups corresponding to compositions of r with n parts. The elements of BL are just the Z-linear endomorphisms of AI which are also right A-module maps.

Notice that BL ∼ = i,j∈I HomA (Ai , Aj ), so bases for each HomA (Ai , Aj ) will combine to give a basis for BL . An element f ∈ HomA (Ai , Aj ) is completely determined by y = f (Gi ) ∈ Aj where Gi = Gi e generates the principal A-module Ai . In fact, f (Gi x) = yx for any x ∈ A. Any y ∈ Aj will define an f ∈ HomA (Ai , Aj ) provided yg = y for all g ∈ Gi . Definition 5.2. R(i, j) = {y ∈ Aj : (∀g ∈ Gi ) yg = y}, the set of elements in Aj which are invariant under the right action of Gi . For any y ∈ R(i, j), write fy for the unique homomorphism in HomA (Ai , Aj ) such that fy (Gi ) = y. Then the correspondence y ↔ fy gives a Z-module isomorphism between R(i, j) and HomA (Ai , Aj ). Any double coset D ∈ j Mi is a disjoint union of left cosets C ∈ j M . Let O(D) = {C ∈ j M : C ⊆ D}. O(D) can be thought of as the orbit of any particular C ⊆ D under the right action of Gi . Define x(D) =  j C∈O(D) C ∈ A . Since right multiplying by a g ∈ Gi just permutes the left cosets in O(D), it leaves x(D) invariant. So x(D) ∈ R(i, j). We will write fD = fx(D) ∈ HomA (Ai , Aj ). Proposition 5.1. (a) R(i, j) is a free Z-module with basis {x(D): D ∈ j Mi }. (b) HomA (Ai , Aj ) is a free Z-module with basis {fD : D ∈ j Mi }. Proof. Since y ↔ fy gives a Z-module isomorphism R(i, j) ∼ = HomA (Ai , Aj ), (a) and (b) are clearly equivalent. We prove (a). Since R(i, j) is a submodule of the free Z-module Aj and each Aj -basis element C ∈ j M occurs in exactly one x(D), the elements {x(D): D ∈ j Mi } are independent. To show that they span R(i, j),  write any x ∈ R(i, j) as x = C∈j M aC C. Suppose C1 , C2 ∈ j M are in the same double coset D ∈ j Mi , so C2 = C1 σ for some σ ∈ Gi . Then since Gi acts on the right on Aj by permuting the basis elements C ∈ j M , the coefficient of C2 = C1 σ in xσ will be aC1 . On the other hand, since x ∈ R(i, j) we have xσ = x,

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so the coefficient of C2 must be aC2 . So the coefficients of left cosets in the same double coset are equal. Write aD = aC where C is any left coset in the double coset D. Then 

x=

 

aC C =

C∈jM

aC C =

D∈jM i C∈D

 D∈jM i

aD



C=

C∈D



aD x(D),

D∈jM i

so the elements {x(D): D ∈ j Mi } span R(i, j) and are a basis. 2

i j It follows from Proposition 5.1, that BL ∼ = i,j∈I HomA (A , A ) is a free Z-module with basis {fD = fx(D) : D ∈ j Mi , i, j ∈ I}. BL is an associative Z-algebra with unit, where the product is composition

of functions. (The unit in BL is just the sum i 1i of the identity mappings 1i : Ai → Ai ∈ HomA (Ai , Ai ) which is clearly in BL .) BL satisfies the following “orthogonality condition”: Let D1 ∈ i Mj , D2 ∈ k Ml , so fD1 : Aj → Ai , fD2 : Al → Ak . Then fD1 fD2 = 0 if k = j, while fD1 fD2 : Al → Ai if k = j. So when k = j  we must have fD1 fD2 = D∈i Ml a(D1 , D2 , D)fD for some structure constants a(D1 , D2 , D) ∈ Z. For any double coset D ∈ j Mi and left coset C ∈ j M such that C ⊆ D, recall that nL (D, i) = o(Gi,C ), the order of the subgroup Gi,C = {σ ∈ Gi : Cσ = C} of Gi . For any m ∈ C ⊆ D, it is easy to check that Gi,C = {σ ∈ Gi : mσ = πm for some π ∈ Gj }. Then, since any coset of Gi,C in Gi contains nL (D, i) elements, a simple computation gives the following lemma. Lemma 5.1. For any C ∈ j M such that C ⊆ D, CX(Gi ) = nL (D, i)x(D) where X(Gi ) =

 σ∈Gi

σ.

We can now calculate the structure constants for BL . Proposition 5.2. Take D1 ∈ i Mj , D2 ∈ k Ml . Then fD1 fD2 = 0 if j = k, while for j = k, fD1 fD2 =

 D∈iM l

nL (D, l) N (D1 , D2 , D)fD . nL (D1 , j)nL (D2 , l)

Proof. When j = k, fD1 fD2 = 0 by definition, so assume j = k. Choose m ∈ D1 , n ∈ D2 and put C1 = Gi m, C2 = Gj n. Then use Lemma 5.1 to compute nL (D1 , j)nL (D2 , l)fD1 fD2 (Gl )   = nL (D1 , j)nL (D2 , l)fD1 x(D2 ) = nL (D1 , j)fD1 C2 X(Gl ) = nL (D1 , j)fD1 (Gj )nX(Gl ) = nL (D1 , j)x(D1 )nX(Gl )  Gi (mρn)X(Gl ). = C1 X(Gj )nX(Gl ) = Gi mX(Gj )nX(Gl ) = ρ∈Gj

Write C(mρn) for the left Gi coset containing mρn and D(mρn) for the double coset in i Ml containing mρn. Then using Lemma 5.1 again gives

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 ρ∈Gj

=

9

Gi (mρn)X(Gl )  ρ∈Gj

C(mρn)X(Gl ) =



    nL D(mρn), l x D(mρn) = nL D(mρn), l fD(mρn) (Gl ).

ρ∈Gj

ρ∈Gj

Recalling that N (D1 , D2 , D) = #{ρ ∈ Gj : D(mρn) = D}, we can rewrite the last sum as × N (D1 , D2 , D)fD (Gl ). Since elements of HomA (Al , Ai ) are determined by their values on Gl , we get nL (D1 , j)nL (D2 , l)fD1 fD2 =



 D∈i Ml

nL (D, l)

nL (D, l)N (D1 , D2 , D)fD .

D∈i Ml

But also nL (D1 , j)nL (D2 , l)fD1 fD2 = nL (D1 , j)nL (D2 , l)



a(D1 , D2 , D)fD

D∈i Ml

where we expand the product fD1 fD2 ∈ HomA (Al , Ai ) in terms of the basis {fD : D ∈ i Ml } with some structure constants a(D1 , D2 , D) that are necessarily integers. Then equating coefficients of basis elements we see that nL (D1 , j)nL (D2 , l) must divide each of the coefficients nL (D, l)N (D1 , D2 , D) and that the (D1 ,D2 ,D) structure constants are, as claimed in the proposition, given by a(D1 , D2 , D) = nnLL(D,l)N (D1 ,j)nL (D2 ,l) . 2 (D1 ,D2 ,D) Note that the structure constants a(D1 , D2 , D) = nnLL(D,l)N (D1 ,j)nL (D2 ,l) , necessarily integers in a Z-algebra, are identical with the aL (D1 , i, j, D2 , j, k, D) of Lemma 4.3. This completes the proof of Lemma 4.3 (and Theorem 4.1).

Corollary 5.1. The correspondence bL (D, j, i) ↔ fD for D ∈ j Mi gives an isomorphism AZI,L ∼ = BL ≡  I ∼ Z ∼ LGS Z (M, G). The identity in LGS Z (M, G) is i∈I fGi . The standard DCA A = C⊗AI,L = LGS C (M, G). Proof. Since the structure constants are identical, the algebras are isomorphic. The identity  for AZI,L must correspond to the identity i∈I fGi for BL . 2

 i∈I

bL (Gi , i, i)

To prove Lemma 4.6 (and Theorem 4.2), one essentially repeats the above argument interchanging “left” and “right”. We omit the parallel proofs noting only the following differences. The map ψ : A → EndZ (AI ) is now an algebra homomorphism, not antihomomorphism, so we regard A as a subalgebra of End Z (AI ) and define B ≡ BR as the “commuting algebra” of A. This algebra BR does not have the “orthogonality op does. Our right Z-form turns out to be isomorphic to this opposite property”, but the opposite algebra BR op , which we define to be the right generalized Schur algebra. (Of course taking the opposite algebra BR algebra is equivalent to “writing endomorphisms on the right” so that the product f ∗ g means “apply f first, then g”.) Definition 5.3. B ≡ BR ≡ End A (AI ). The right generalized Schur algebra over Z, RGS Z (M, G), is the op ≡ (End A (AI ))op . For any domain R, the right generalized Schur algebra over R is opposite Z-algebra BR the R-algebra RGS R (M, G) = R ⊗Z RGS Z (M, G). The elements of BR are now the Z-linear endomorphisms of AI which are also left A-module maps. op op . Then the structure constants for BR are given by the Write ∗ for the product in the opposite algebra BR following analog to Proposition 5.2:

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Proposition 5.3. Take D1 ∈ i Mj , D2 ∈ k Ml . Then fD1 ∗ fD2 = 0 if j = k, while if j = k, 

fD1 ∗ fD2 =

D∈i Ml

nR (D, i) N (D1 , D2 , D)fD . nR (D1 , i)nR (D2 , j)

(D1 ,D2 ,D) This time the structure constants a(D1 , D2 , D) = nnRR(D,i)N (D1 ,i)nR (D2 ,j) (which are necessarily integers) are identical with the aR (D1 , i, j, D2 , j, k, D) of Lemma 4.6. This completes the proof of Lemma 4.6 (and Theorem 4.2). op Corollary 5.2. The correspondence bR (D, i, j) ↔ fD gives an isomorphism AZI,R ∼ = BR ≡ RGS Z (M, G).  I ∼ Z The identity in RGS Z (M, G) is i∈I fGi . The standard DCA A = C ⊗ AI,R ∼ = RGS C (M, G).

It is possible to repeat all of the above constructions using any commutative domain R with identity in place of Z, obtaining an algebra A(R) = R[M ], R-modules AI (R), and R-algebras BL (R) ≡ LGS(R, M, G), op BR (R) ≡ RGS(R, M, G). Evidently A(R) ∼ = R ⊗Z A ≡ R ⊗Z Z[M ]. With this identification it is not hard to check that the isomorphism AI (R) ∼ = R ⊗Z AI of free R-modules is actually an A(R)-module isomorphism. op op Similarly, the isomorphisms BL (R) ∼ = R ⊗Z BL ≡ LGS R (M, G), BR (R) ∼ = R ⊗Z BR ≡ RGS R (M, G) of free R-modules are actually isomorphisms of R-algebras, giving direct constructions of the left and right op generalized Schur algebras. The structure constants for BL (R) and BR (R) are given by ψ(aL ) and ψ(aR ) Z Z where aL , aR are the structure constants for the Z-forms AI,L , AI,R and ψ : Z → R is the natural ring homomorphism taking 1 → 1. When R = C (or any ring of characteristic 0), ψ is injective and identifies Z with a subring of R, so the structure constants are just aL , aR . There are then strings of isomorphisms LGS(C, M, G) ≡ BL (C) ∼ = C ⊗Z BL ≡ LGS C (M, G) ∼ = C ⊗Z AZI,L ∼ = AI and op op RGS(C, M, G) ≡ BR (C) ∼ = C ⊗Z BR ≡ RGS C (M, G) ∼ = C ⊗Z AZI,R ∼ = AI .

We remark that it is sometimes useful to view both of the Z-forms AZI,L , AZI,R as just the free Z-module with a basis MI of all double cosets equipped with new “left” and “right” products ∗L and ∗R . So for D1 ∈ i Mj , D2 ∈ k Ml , D1 ∗L D2 = D1 ∗R D2 = 0 when j = k, while if j = k, D1 ∗L D2 =

 D∈l Mi

nL (D, l) N (D1 , D2 , D)D nL (D1 , j)nL (D2 , l)

and D1 ∗R D2 =

 D∈i Ml

nR (D, i) N (D1 , D2 , D)D. nR (D1 , i)nR (D2 , j)

A monoid isomorphism h : M → M  takes a family G = {Gi : i ∈ I} of subgroups of M to a family G = {Gi : i ∈ I} of subgroups of M  where Gi = h(Gi ). Evidently, h induces isomorphisms between corresponding DCAs, left standard Z-forms, left generalized Schur algebras, etc. A monoid anti-isomorphism h : M → M  also takes the family G to the family G and induces an anti-isomorphism between corresponding DCAs. h takes a double coset D ∈ i Mj to a double coset D ∈ j Mi . (Note the reversal of indices.) This results in an anti-isomorphism of the left standard Z-form for M with the right standard Z-form for M  . Similarly, the left generalized Schur algebra corresponding to M is anti-isomorphic to the right generalized Schur algebra corresponding to M  . Applying this to a monoid with an anti-automorphism gives the following result.

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Proposition 5.4. Suppose the monoid M admits an anti-automorphism h : M → M such that h(Gi ) = Gi for all i ∈ I. Then h yields (a) an anti-automorphism hS : AI → AI of the standard DCA, hL

(b) anti-isomorphisms AZI,L  AZI,R between left and right standard Z-forms, hR

hL

(c) anti-isomorphisms LGS R (M, G)  RGS R (M, G) between left and right generalized Schur algebras, hR

(d) an isomorphism hB : BL → BR , and (e) isomorphisms hB : BL (R) → BR (R) for any domain R. So for monoids satisfying the hypotheses of the proposition, the left and right standard Z-forms and the left and right generalized Schur algebras agree up to anti-isomorphism, while the endomorphism algebras BL , BR are isomorphic. Notice that if M is a group G, then h : g → g −1 always yields an anti-automorphism with h(Gi ) = Gi for all i ∈ I, so the proposition applies. Similarly, if M = , the rook algebra of r by r matrices with entries in {0, 1} and at most one 1 in each row and each column, then h : m → mT , where mT is the transpose matrix, is an anti-automorphism of M . If m is an invertible element, then h(m) = mT = m−1 , so h(Gi ) = Gi for any subgroup of M and the proposition applies. 6. Examples: Hecke algebras When G = {G} consists of a single subgroup G of the monoid M , the standard DCA AI ≡ AG is isomorphic to εC[M ]ε where ε = X(G) o(G) is an idempotent in the monoid algebra C[M ]. The algebra εC[M ]ε is the Iwahori–Hecke algebra H(M, G) of M relative to G (see [1,7,6], and [2]). Theorems 4.1 and 4.2 then provide Z-forms AZL and AZR for any Hecke algebra H(M, G) ∼ = AI . The existence of a Z-form for the Hecke algebra corresponding to any (finite) monoid M appears to be a new result. When M itself is a group, any left or right G-coset will contain o(G) elements and, in the notation of Section 3, nL (D) = nR (D) = o(G). Then the rescaled bases for the left and right Z-forms are idenZ Z tical: bL,D = bR,D = X(D) o(G) . So the left and right Z-forms AL and AR are identical (and agree with the “well-known” Z-form for this case). The structure constants are given by aL (D1 , D2 , D) = aR (D1 , D2 , D) =

n(D)N (D1 , D2 , D) n(D1 )n(D2 )

where, for D = GmG, n(D) = |{(σ, π) ∈ G × G: σmπ −1 = m}| is the order of the subgroup of G × G which leaves an element m ∈ D fixed. For M = GL(n, Fq ), the monoid (group) of n by n nonsingular matrices over a finite field Fq with q elements, and G = D the group of upper triangular matrices in M , H(M, G) is the standard Iwahori–Hecke algebra (see for example [3] or [7]). We sketch how this well-known case fits with our development. The Bruhat decomposition shows that the double cosets are indexed by permutation matrices σ ∈ Sn : For any double coset D there is a unique σ ∈ Sn such that D = Dσ = GσG. A little computation gives (n−1)n o(G) = q 2 (q − 1)n and a longer argument gives n(Dσ ) = n(GσG) = o(G)/q l(σ) where l(σ) is the length of the permutation σ ∈ Sn . Let s be an elementary transposition in Sn . A study of the Bruhat decomposition yields N (Ds , Dσ , D) = 0 if D = Dσ , Dsσ ; if l(sσ) = l(σ) + 1, then N (Ds , Dσ , Dσ ) = 0, N (Ds , Dσ , Dsσ ) = o(G); if l(sσ) = l(σ) − 1, then N (Ds , Dσ , Dσ ) = (q−1)o(G) , N (Ds , Dσ , Dsσ ) = o(G) q q . A computation of the structure constants then gives the usual formulas for multiplication in the standard Iwahori–Hecke algebra:  Ds Dσ =

Dsσ qDsσ + (q − 1)Dσ

if l(sσ) = l(σ) + 1 if l(sσ) = l(σ) − 1.

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In this case, it can be checked (by induction on the length of σ1 ) that all structure constants for q ) ∈ Z[q] H(GL(n, Fq ), D) are “universal polynomials” in q. That is, there are polynomials a(Dσ1 , Dσ2 , Dσ )(ˆ such that the evaluation at q, a(Dσ1 , Dσ2 , Dσ )(q), is the corresponding structure constant for H(Mn (Fq ), D). This suggests the construction of the “generic Iwahori–Hecke algebra” Hq : Form the free Z[q]-module with basis {Tσ : σ ∈ Sn }. Define a Z[q]-bilinear product by Tσ1 Tσ2 = a(Dσ1 , Dσ2 , Dσ )(q)Tσ . This product will be associative if it satisfies the associative condition of Section 4. This associative condition amounts to the vanishing of certain polynomials in Z[q]. These polynomials must vanish whenever we evaluate at q = q = pk , a prime power, for their vanishing is just the associative condition for the associative algebra H(Mn (Fq ), D). So the polynomials vanish identically and the generic algebra Hq is an associative Z[q] algebra (with unit Te where e is the unit in Sn ). For any domain R and element q ∈ R we can then define the Iwahori–Hecke algebra HR,q = R ⊗Z[ˆq] Hqˆ where we view R as a right Z[q]-module via the homomorphism ψ : Z[q] → R where ψ(1) = 1, ψ(q) = q. Our original algebra H(GL(n, Fq ), D) is then isomorphic to HC,q . When the monoid M is not a group, the left and right Z-forms for a Hecke algebra H(M, G) ∼ = AI are in general different. They also differ in general from the Z-form for the Hecke algebra associated with a Renner monoid as in [2]. For example, in the case considered by Solomon in [7], Mn is the monoid of all n by n matrices over the finite field Fq with q elements and G ⊆ Mn is the group of nonsingular, upper triangular matrices. In this situation a Bruhat type decomposition shows that the double G–G cosets are indexed by elements of the rook monoid, : for any double coset D there exists a unique element α ∈  such that D = Dα = GαG. For this case, the structure constants for H(M, G) with respect to various relevant bases can be expressed in terms of length functions on . We sketch the results. For α ∈ , define the left length by ll(α) = minimum number of row transpositions required to bring the rows of α into standard order: all zero rows at the bottom of the matrix; for nonzero rows i and j, row i is above row j if the 1 in row i is to the left of the 1 in row j. Similarly define the right length by rl(α) = minimum number of column transpositions required to bring the columns of α to standard order: all zero columns to the left of the matrix; for nonzero columns i and j, column i is to the left of column j if the 1 in column i is above the 1 in column j. Finally, when α has rank r, the length of α (as defined by Solomon in [7]) can be defined as l(α) = minimum number of row and column transpositions required to bring α to standard form: all zero rows at the bottom, all zero columns at the left, and the r by r submatrix lying in the first r rows and last r columns equal to the r by r identity matrix. For the left standard Z-form, one finds nL (Dα , G) = qo(G) rl(α) and then by Lemmas 4.1 and 4.2 the scale factor is rL (Dα ) =

n(Dα )q rl(α) (o(G))2 .

The structure constants are

aL (Dα1 , Dα2 , Dα ) =

q rl(α1 ) q rl(α2 ) N (Dα1 , Dα2 , Dα ) nL (Dα , G)N (Dα1 , Dα2 , Dα ) = . nL (Dα1 , G)nL (Dα2 , G) q rl(α) o(G)

When s ∈ Sn ⊆  is a basic transposition, one can then compute ⎧ ⎨ Dαs Dα Ds = qDαs = qDα ⎩ qDαs + (q − 1)Dα

if rl(αs) = rl(α) + 1 if rl(αs) = rl(α) if rl(αs) = rl(α) − 1.

(1)

Next let Δ ∈  be the matrix of rank n − 1 with 1’s in row i, column i + 1, for 1  i  n − 1 and zeros elsewhere. Then for any α ∈ , rl(αΔ)  rl(α) and one finds Dα DΔ = q rl(α)−rl(αΔ) DαΔ

(2)

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One can then show that any Dβ can be written as a product of various Ds and DΔ . Then the general structure constants for Dα Dβ are determined by our special cases. By induction on the number of such basic factors in Dβ , one finds that the structure constants (already known to be integers) are all universal polynomials in qˆ evaluated at qˆ = q. So we can construct a “generic Iwahori–Hecke algebra” Hqˆ,L (), a free Z[ˆ q ]-module with basis {Tσ : σ ∈ } and structure constants aL as given above. Then there are isomorphisms AZI,L ∼ = AI ∼ = Z ⊗Z[ˆq] Hqˆ,L () and H(M, G) ∼ = C ⊗Z AZI,L ∼ = C ⊗Z[ˆq] Hqˆ,L (). Similarly, for the right standard Z-form, one finds nR (Dα , G) = qo(G) ll(α) and then by Lemmas 4.4 and 4.5 the scale factor is rR (Dα ) =

n(Dα )q ll(α) (o(G))2 .

aR (Dα1 , Dα2 , Dα ) =

The structure constants are now

q ll(α1 ) q ll(α2 ) N (Dα1 , Dα2 , Dα ) nR (Dα , G)N (Dα1 , Dα2 , Dα ) . = nR (Dα1 , G)nR (Dα2 , G) q ll(α) o(G)

When s ∈ Sn ⊆  is a basic transposition, the analog of Eq. (1) is now ⎧ ⎨ Dsα Ds Dα = qDsα = qDα ⎩ qDsα + (q − 1)Dα

if ll(sα) = ll(α) + 1 if ll(sα) = ll(α) if ll(sα) = ll(α) − 1.

(3)

For any α ∈ , ll(Δα)  ll(α) and one finds DΔ Dα = q ll(α)−ll(Δα) DΔα

(4)

Notice the reversal of order in these formulas from Eqs. (1) and (2). Eqs. (1) and (2) (with rl replaced by ll) are in general not satisfied. One can again show that any Dβ can be written as a product of various Ds and DΔ , so the general structure constants for Dα Dβ are again determined by our special cases. The structure constants (integers) are again all certain universal polynomials in qˆ evaluated at qˆ = q. So we can construct another “generic q ]-module with basis {Tσ : σ ∈ } and structure constants aR as Iwahori–Hecke algebra” Hqˆ,R (), a free Z[ˆ given above. Then there are isomorphisms AZI,R ∼ = AI ∼ = Z ⊗Z[ˆq] Hqˆ,R () and H(M, G) ∼ = C ⊗Z AZI,R ∼ = C ⊗Z[ˆq] Hqˆ,R (). Finally, if one uses Solomon’s length function, l, an appropriate rescaling is given by rS (Dα ) = q l(α1 ) q l(α2 ) N (D

,D

n(Dα )q l(α) (o(G))2 .

,D )

α1 α2 α . For Solomon’s length function, The structure constants are aS (Dα1 , Dα2 , Dα ) = q l(α) o(G) Eqs. (1), (2), (3), and (4) are all satisfied (with rl and ll replaced by l). Then one can again show that any Dβ can be written as a product of various Ds and DΔ and the general structure constants for Dα Dβ are again determined by the special cases. By induction on the lengths, the structure constants are again found to be integers so we have a new Z-form AZS for AI . The structure constants are again all certain universal polynomials in qˆ evaluated at qˆ = q, so we can construct the “usual” q ]-module with basis generic Iwahori–Hecke algebra Hqˆ() known as the q-rook algebra. This is a free Z[ˆ {Tσ : σ ∈ } and structure constants aS as given above. Then there are isomorphisms AZS ∼ = Z ⊗Z[ˆq] Hqˆ() Z ∼ ∼ ∼ and H(M, G) = AI = C ⊗Z AS = C ⊗Z[ˆq] Hqˆ(). We conjecture that for any of the Hecke algebras associated with Renner monoids M considered by Godelle [2] our left and right Z-forms may be defined in terms of some left and right length functions defined on M .

7. Examples: Schur algebras In this section we justify the term “generalized Schur algebra” by showing that when M is the symmetric group Sr and G is the collection of “Young subgroups” Sλ corresponding to compositions λ of r in n

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parts, then the left generalized Schur algebra BL (R) ≡ LGS R (M, G) is isomorphic to the usual Schur algebra SR (n, r) over the domain R. (In some treatments, the classical Schur algebra is taken to be the opposite algebra of this SR (n, r), in which case by Proposition 5.4 it is isomorphic to the right generalized Schur algebra BR (R)op ≡ RGS R (M, G).) When the symmetric group is replaced by the full transformation semigroup τr (or one of the related monoids in [5,4]), an argument similar to that below shows that the right generalized Schur algebras defined here are isomorphic to the generalized Schur algebras of [5] and [4]. Let V be an n-dimensional C vector space with basis {b1 , b2 , . . . , bn } and let V r = V ⊗ V ⊗ · · · ⊗ V be the product of r copies of V . Then V r has a basis b(V r ) = {bi1 ⊗ bi2 ⊗ · · · ⊗ bir : ik ∈ n ¯ } where n ¯ = {1, 2, . . . , n}. Write h(r, n) for the set of maps r¯ → n ¯ and for any α ∈ h(r, n) define bα = bα(1) ⊗ bα(2) ⊗ · · · ⊗ bα(r) . Then we can write the basis as b(V r ) = {bα : α ∈ h(r, n)}. The symmetric group Sr acts on V r by “permuting the factors”. More precisely, a right action of Sr is defined on basis elements by (bα )σ = bασ . The classical Schur algebra S(n, r) can be defined as the subalgebra of EndC (V r ) consisting of endomorphisms which commute with the action of Sr : S(n, r) = End C[Sr ] (V r ). We will show that S(n, r) ∼ = LGS C (M, G) ∼ = BL (C) for a certain family G = {Gi : i ∈ I} of subgroups of Sr . Let Λ(r, n) be the set of compositions of r with n parts: if λ = {λi : i ∈ n ¯ } ∈ Λ(r, n), then 0  λi and   λ = r. For each λ ∈ Λ(r, n) let S be the corresponding “Young subgroup” of Sr , Sλ ∼ = i∈¯n Sλi . λ i∈¯ n i Define a set of basis vectors bλ (V r ) = {bα : |α−1 (i)| = λi for all i ∈ n ¯ }, that is, the set of all products ¯ ∈ h(r, n) to bα = bα(1) ⊗ bα(2) ⊗ · · · ⊗ bα(r) where each basis element bi appears exactly λi times. Define λ ¯ = k for λ1 + λ2 + · · · + λk−1 < i  λ1 + λ2 + · · · + λk−1 + λk . Then b ¯ ∈ bλ (V r ) and be the map with λ(i) λ r bλ (V ) = bλ¯ · Sr . Also, bλ¯ σ = bλ¯ ⇔ σ ∈ Sλ , so the elements of bλ (V r ) correspond one to one with the left cosets Sλ ρ, ρ ∈ Sr . Furthermore, the right action of Sr on the basis elements bλ (V r ) corresponds to the right action on the cosets Sλ ρ, ρ ∈ Sr . Write V λ for the subspace of V r spanned by bλ (V r ) and notice

that V r = λ∈Λ(r,n) V λ . Now let I index the collection {Sλ : λ ∈ Λ(r, n)} of subgroups of the monoid Sr and construct the left coset modules Aλ = Ai and AI as in Section 5. Then we have right Sr -module isomorphisms V λ ∼ = Aλ and V r ∼ = End C[Sr ] (AI ) ≡ LGS C (Sr , G) ∼ = BL (C) ∼ = AI . Evidently = AI . Then S(n, r) = End C[Sr ] (V r ) ∼ BL = BL (Z) serves as a Z-form for the classical Schur algebra S(n, r) and then for any domain R we have SR (n, r) ∼ = R ⊗ BL ∼ = LGS R (Sr , G) as claimed. The generalized Schur algebras of [5] and [4] can also be thought of as double coset algebras. Let τ¯r be the monoid of all maps α : {0, 1, . . . , r} → {0, 1, . . . , r} which fix 0, α(0) = 0. Then τ¯r can be considered to contain the symmetric group Sr and the full transformation semigroup τr . If S is any submonoid of τ¯r which contains Sr and R is any commutative domain, then a generalized Schur algebra denoted by B(S, R) is constructed in [4]. It is not hard to check that this B(S, R) is isomorphic to the right generalized Schur op algebra RGS R (M, G) ≡ BR (R)op ∼ = R ⊗ BR of Section 5 where the monoid M = S and I indexes the collection {Sλ : λ ∈ Λ(r, n)} of subgroups of M . As a final remark, we note that the q-Schur algebras Sq (n, r) can also be constructed in the context of double coset algebras. Let M be the group of nonsingular r by r matrices over a finite field with q elements and consider the symmetric group Sr as a subgroup of M . For each λ ∈ Λ(r, n), let Gλ be the subgroup of M generated by Sλ ∪ U where U is the group of nonsingular upper triangular matrices. It can be checked that Gλ consists of those elements of M of the form U1 σU2 for U1 , U2 ∈ U , σ ∈ Sλ , that is, those elements whose Bruhat decomposition involves a σ ∈ Sλ . Right cosets of Gλ in M correspond one to one with right cosets of Sλ in Sr , and double cosets Gλ mGμ in M correspond one to one with double cosets Sλ σSμ in Sr . The right coset endomorphism algebra BR (C) corresponding to the monoid M and the collection of subgroups {Gλ : λ ∈ Λ(n, r)} can then be identified with the q-Schur algebra Sq (n, r) over C (and BR is a Z-form for this algebra). It can be checked that the structure constants for the corresponding DCA are given by a universal polynomial in q, so a “generic” q-Schur algebra Sqˆ(n, r) can be defined over the domain Z[ˆ q ] as for the Hecke algebras. Then for any domain R and q ∈ R, we have SR,q (n, r) = R ⊗Z[ˆq] Sqˆ(n, r).

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