A generic algebra associated to certain Hecke algebras

Journal of Algebra 278 (2004) 502–531 www.elsevier.com/locate/jalgebra

A generic algebra associated to certain Hecke algebras ✩ Stephen Doty a,∗ , Karin Erdmann b , Anne Henke c a Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626, USA b Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK c Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, UK

Received 15 May 2003 Available online 7 June 2004 Communicated by Alexander Premet

Abstract We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain generic algebra, infinite-dimensional in general, coming from the universal enveloping algebra of sln (or gln ). The endomorphism algebras and the generic algebras are cellular (in the latter case, of profinite type in the sense of R.M. Green). We give several equivalent descriptions of these algebras, find a number of explicit bases for them, and describe indexing sets for their irreducible representations. Moreover, we show that the generic algebra embeds densely in an endomorphism algebra of a certain infinite-dimensional induced module.  2004 Elsevier Inc. All rights reserved.

Introduction We study the intertwining spaces λ Sµ = λ S(n, r)µ := HomΣr (M µ , M λ ) between permutation modules M µ , M λ for a symmetric group Σr . Here λ, µ are given n-part compositions of r; that is, arbitrary sequences of n nonnegative integers which sum to r. We are particularly interested in the endomorphism algebras S(λ) := λ S(n, r)λ . ✩

The first author was supported at Oxford by an EPSRC Visiting Fellowship; all authors gratefully acknowledge support from Mathematisches Forschungsinstitut Oberwolfach, Research-in-Pairs Program. * Corresponding author. E-mail addresses: [email protected] (S. Doty), [email protected] (K. Erdmann), [email protected] (A. Henke). 0021-8693/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2004.04.007

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

503

Permutation modules are of central interest for the representation theory of symmetric groups, and moreover provide a natural link with the representation theory of general linear groups, via Schur algebras. Given a fixed n-dimensional vector space E over a field K, the symmetric group Σr acts on the right on E⊗r by place permutations, and the endomorphism
algebra S(n, r) = EndΣr (E⊗r ) is a Schur algebra. As a KΣr -module, E⊗r = λ M λ (sum over all n-part compositions of r), and thus S(n, r) =



λ S(n, r)µ

λ,µ

=



1λ S(n, r)1µ

(a)

λ,µ

where 1λ ∈ S(n, r) is the idempotent corresponding to the projection E⊗r → M λ . Each intertwining space λ S(n, r)µ is naturally an S(λ)-S(µ) bimodule, and the algebras S(λ) appear ‘on the diagonal’ in this decomposition. We expect that a systematic study of these algebras and bimodules will ultimately lead to a better understanding of Schur algebras; however, the finite-dimensional algebras S(λ) are interesting in their own right. Moreover, the group algebra of Σr appears in this theory as the special case S(ω) = ω S(r, r)ω for the particular partition ω = (1r ), so this extends the study of symmetric groups to a broader context. Our interest in the S(λ) is motivated by the fact that knowing the dimensions of the simple S(λ)-modules for all λ is equivalent to knowing the characters of the simple polynomial GLn (K)-modules for K infinite; see Section 4. In Proposition 3.5 we show that λ Sµ identifies with HomMn (K[Mn ]µ , K[Mn ]λ ); in particular, the algebra S(λ) identifies with EndMn (K[Mn ]λ ). Proposition 3.5 was already known for the case n
r; we extend this (with a new proof) to the general case. As an application, we obtain a new characteristic-free proof of “Schur–Weyl duality,” which gives a new proof of a theorem of de Concini and Procesi (see 3.6). We also describe several natural bases for the spaces λ Sµ , the least elementary of which is the PBW-type basis in Proposition 5.5, an application of results of [11]. Our main objects of study are certain infinite-dimensional analogues of the algebras S(λ), and of the bimodules λ Sµ . These occur naturally in Lusztig’s construction of the modified form U˙ of a universal enveloping algebra U corresponding to a reductive Lie algebra. The modified form U˙ is obtained by replacing the zero part of U by the direct sum of infinitely many one-dimensional algebras (and thus U˙ has an infinite family of ˙ n ) has orthogonal idempotents) indexed by the weight lattice. In particular, the algebra U(gl a decomposition ˙ n) = U(gl



˙ n )1µ 1λ U(gl

(b)

λ,µ∈Zn

˙ ˙ n )1λ appear on analogous to the decomposition (a). The generic algebras U(λ) := 1λ U(gl ˙ ˙ ˙ the diagonal in the decomposition, and the spaces 1λ U(gln )1µ are U(λ)-U(µ) bimodules. n ˙ We have for any λ ∈ N natural quotient maps U(λ) → S(λ) (Proposition 9.4), and there ˙ is an explicit PBW-type basis of U(λ) compatible with the PBW-type basis of S(λ) (Proposition 7.3). These quotient maps are also compatible with the canonical basis and cell structure (Theorem 11.8).

504

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

There is a surjective map λ → λ˜ taking elements of Zn to elements of Zn−1 , given by the rule (λ1 , . . . , λn ) → (λ1 − λ2 , . . . , λn−1 − λn ). For any λ, µ ∈ Zn we have an ˙ n )1µ  1 ˜ U(sl ˙ n )1µ˜ (see Proposition 10.3). In particular, the generic isomorphism 1λ U(gl λ algebra depends up to isomorphism only on λ˜ and not λ. The decomposition  ˙ n )1µ˜ ˙ n) = 1λ˜ U(sl (c) U(sl ˜ µ∈Z λ, ˜ n−1

has just one copy of each generic algebra, whereas the decomposition (b) has infinitely many. All algebras S(λ) for compositions λ belonging to the fiber of λ˜ are quotients of the ˙ λ), ˜ coming from the decomposition (c); see 10.5. same generic algebra U( ˙ λ) ˜ as an endomorphism algebra of an appropriate One would like an interpretation of U( induced module, but we were not able to obtain such an interpretation. Instead, we show that it embeds densely in the algebra EndSLn (K[SLn ]λ ) (Corollary 12.7). For this we need to consider a certain completion  U(sln ) obtained as an inverse limit of an appropriate inverse system of Schur algebras. Although q-analogues of all these connections obviously exist, we do not address them here, in the interest of keeping the paper to a reasonable length.

1. Induction and weight spaces 1.1. Fix an infinite field K. We consider the algebraic monoid Mn (K) of n × n matrices over K and its unit group GLn (K), along with the subgroup SLn (K) of elements of determinant 1. Let M be one of Mn , GLn , or SLn . In all three cases M is an affine algebraic monoid determined by its corresponding bialgebra K[M]. (General references for the properties of algebraic monoids are [28,30].) Let D be the submonoid of diagonal matrices in M and X(D) the monoid of characters on D. We have identifications (N always denotes the set of nonnegative integers) X(D)  Nn , Zn , Zn−1

(respectively).

(1.1.1)

There is on K[M] a natural rational bimodule structure, with M acting on the left via (m, f ) → Rm f and on the right via (f, m) → Lm f , for all m ∈ M, f ∈ K[M], where (Lm f )x = f (mx),

(Rm f )x = f (xm) (x ∈ M).

Let λ K[M] (respectively, K[M]λ ) be the λ-eigenspace for the left (respectively, right) action of D on K[M] obtained by restricting the M action. We have direct sum decompositions   K[M]λ . (1.1.2) K[M] = λ K[M] = λ∈X(D)

λ∈X(D)

We note that λ K[M] is a rational right M-module and K[M]λ a rational left one, since L, R commute. In fact,

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

505

  K[M]λ = f ∈ K[M]: Ld f = λ(d)f, all d ∈ D   = f ∈ K[M]: f (dx) = λ(d)f (x), all d ∈ D, x ∈ M . Similarly, λ K[M] =

  f ∈ K[M]: f (xd) = λ(d)f (x), all d ∈ D, x ∈ M .

In other words, we have isomorphisms M λ K[M]  indD

K[M]λ  indM D λK

Kλ ,

(1.1.3)

of rational right (respectively, left) M-modules, where Kλ (respectively, λ K) is the onedimensional right (respectively, left) D-module affording the character λ. (Our definition of induced module is not the usual one, e.g., [23], used for algebraic groups, since in the monoid case one cannot twist by the inverse anti-automorphism. For details on the properties of induced modules for affine algebraic monoids, see [9] or [8].) Note that from Frobenius reciprocity it is immediate that induction takes injectives to injectives; hence K[M]λ and λ K[M] are injective as rational M-modules. Later we shall need the finer decomposition K[M] =



λ K[M]µ

(1.1.4)

λ,µ∈X(D)

where λ K[M]µ := λ K[M] ∩ K[M]µ . Also for later use we record the following isomorphisms, which follow immediately by Frobenius reciprocity:   K[M]∗λ  HomM K[M], λ K[M] ,   ∗ µ K[M]  HomM K[M], K[M]µ ,     ∗ λ K[M]µ  HomM λ K[M], µ K[M]  HomM K[M]µ , K[M]λ

(1.1.5) (1.1.6) (1.1.7)

for any λ, µ ∈ X(D). 1.2. The modules K[GLn ]λ and λ K[GLn ] (any λ ∈ Zn ) are infinite-dimensional; similarly for K[SLn ]λ˜ and λ˜ K[SLn ] (any λ˜ ∈ Zn−1 ). On the other hand, K[Mn ]λ and n λ K[Mn ] (any λ ∈ N ) are finite-dimensional. The finite-dimensionality follows from Lemma 1.3 below, while it is well-known that nonzero rational injective G-modules are infinite-dimensional (cf. [23, II, 11.17 Proposition]) for G any affine algebraic group. Let cij be the element of K[Mn ] given by evaluation of a matrix at its (i, j )th entry. Since K is infinite, the cij are algebraically independent. Thus we may identify K[Mn ] with the polynomial algebra K[cij ] in the n2 variables cij (1  i, j  n). The algebra K[GLn ] may be identified with the localization of K[cij ] at  the element det := det(cij ). The comultiplication on K[GLn ] is determined by ∆(cij ) = k cik ⊗ ckj and the counit in terms of Kronecker delta by (cij ) = δij . Note that det is a group-like element. One may regard K[Mn ] as a subalgebra (in fact it is a sub-bialgebra) of K[GLn ] via the

506

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

injection K[Mn ] → K[GLn ] given by restricting functions from Mn to GLn . This gives a categorical equivalence between polynomial GLn -modules and rational Mn -modules. The natural bimodule structure (see 1.1) on K[Mn ] is given explicitly by g · cij =



cij · g =

cik gkj ,

k



gik ckj

(g ∈ Mn ).

(1.2.1)

k

These formulas also give the bimodule structure on K[GLn ], simply by restricting g to GLn . Moreover, by restricting to D = Dn , the submonoid of diagonal elements in Mn , we obtain equalities t · cij = cij tjj = t εj cij ,

cij · t = tii cij = t εi cij

(t ∈ Dn ).

(1.2.2)

This shows that cij belongs to the weight space εj K[Mn ]εi ; i.e., cij has left weight εj and right weight εi , where ε1 , . . . , εn is the standard basis of Zn . We have the following version of [12, (2.7), (2.12)] or [7, proof of (3.4)(i)], which provides an alternative description of the weight spaces in K[Mn ], in terms of generalized symmetric powers of the natural module E, and shows that these weight spaces are finitedimensional. We may regard E = K n as both a left and right module for Mn , with the action given by matrix multiplication of elements of E regarded as column or row vectors as appropriate. Given λ ∈ Nn , set S λ E = (S λ1 E) ⊗ · · · ⊗ (S λn E), which may be regarded as both a left and right Mn -module. Lemma 1.3. Let Dn be the monoid of diagonal elements of Mn . For any λ ∈ Nn there is Mn n λ an isomorphism between indM Dn Kλ (respectively, indDn λ K) and S E, as left (respectively, right) rational Mn -modules. Proof. Let (ei ) be the canonical basis of E. We may identify S a E with homogeneous polynomials in the ei of degree a. The left and right actions of Mn on S a E are by linear substitutions. By (1.2.2) the left weight space λ K[Mn ] is spanned by all monomials of the form

aij  cij aij = λj , for all j (1.3.1) i,j

i

with Mn acting on the right as linear substitutions by the second formula in (1.2.1) above. Via the identifications of (1.1.3), the map taking an element of form (1.3.1) onto the element 





a a a ei i1 ⊗ ei i2 ⊗ · · · ⊗ ei in (1.3.2) i

i

defines the desired isomorphism of right Mn -modules. The other case is similar. 2

i

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

507

2. Schur algebras 2.1. The Schur algebra, SK (n, r) = EndΣr (E⊗r ), provides an approach to the polynomial representation theory of GLn (i.e., rational representation theory of Mn ) through finite-dimensional algebras. Any polynomial GLn -module can be written as a direct sum of its various homogeneous components, and the category of degree r homogeneous GLn modules is equivalent to the category of SK (n, r)-modules. In particular, we have an isomorphism     HomGLn V , V   HomSK (n,r) V , V 

(2.1.1)

for any degree r homogeneous GLn -modules V , V  (see [17, §2]). 2.2. The Schur algebra can alternatively be constructed as the linear dual of the coalgebra AK (n, r), the K-linear span of the monomials in the cij in K[Mn ] of total degree r. This provides a basis for SK (n, r), as follows. Given a pair of multi-indices i, j in I (n, r), define ci,j = ci1 j1 ci2 j2 · · · cir jr .

(2.2.1)

Here, I (n, r) is the set of multi-indices i = (i1 , . . . , ir ), where each ik belongs to {1, . . . , n}. The commutativity of the variables cij implies that we have to take into account the equality rule ci,j = ci ,j

⇔

i = iπ, j = jπ,

for some π ∈ Σr ,

(2.2.2)

with respect to the obvious right action of Σr on I (n, r). As a K-space, AK (n, r)∗ has basis {ξi,j } dual to the basis {ci,j }. There is a similar equality rule for the ξi,j . The distinct ξi,j provide the desired basis for the Schur algebra SK (n, r). In particular, by [17, 3.2] the distinct elements of the form ξi,i for i ∈ I (n, r) provide a set of orthogonal idempotents which add up to the identity in SK (n, r). Given i we set wt(i) = (λ1 , . . . , λn ) ∈ Nn where λj is defined to be the number of ik which equal j . We shall write 1λ := ξi,i



 λ = wt(i) as above .

(2.2.3)

The distinct idempotents in this family are parametrized by the set 

  Λ(n, r) := λ ∈ Nn  λi = r i

(2.2.4)

of n-part
compositions of r. For any left SK (n, r)-module V , one has a decomposition V = λ∈Λ(n,r) 1λ V . For any λ ∈ Λ(n, r) one has by [17, 3.2] an identification 1λ V = λ V

(2.2.5)

508

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

where V is regarded as a polynomial GLn -module (i.e., rational Mn -module) and λ V is the left weight space for Dn of weight λ. A similar statement applies to right modules. It follows from (1.2.1) that AK (n, r), regarded as Mn -Mn bimodule, is r-homogeneous for either action. Moreover, AK (n, r) is an SK (n, r)-SK (n, r) bimodule (see [17, 2.8]). In particular, this means that the right (respectively, left) weight space AK (n, r)λ (respectively, λ AK (n, r)) is a left (respectively, right) SK (n, r)-module. Lemma 2.3. Let λ ∈ Λ(n, r). (a) AK (n, r)λ = K[Mn ]λ and λ AK (n, r) = λ K[Mn ]. (b) We have isomorphisms (AK (n, r)λ )∗  1λ SK (n, r) (as right SK (n, r)-modules) and (λ AK (n, r))∗  SK (n, r)1λ (as left SK (n, r)-modules). Proof. (a) The space K[Mn ]λ is spanned by {cu,i | i ∈ I (n, r)}, where u is a fixed element of I (n, r) of weight λ, so there is an inclusion K[Mn ]λ ⊂ AK (n, r)λ . The opposite inclusion is obvious. This establishes the first equality in (a). The proof of the second equality in (a) is similar. (b) We prove only the first statement, since the other case is similar. The right SK (n, r)module structure on AK (n, r)λ ∗ is induced from the right Mn -action, given by   (ξ · m)c = ξ(mc) ξ ∈ AK (n, r)λ ∗ , m ∈ Mn , c ∈ AK (n, r)λ . The space AK (n, r)λ ∗ can be naturally identified with the span of all ξu,i with u, i as in (a), giving the stated isomorphism. 2 Remark 2.4. It follows from part (b) of the preceding result that we have an isomorphism ∗  (2.4.1) µ AK (n, r)λ  1λ SK (n, r)1µ for any λ, µ ∈ Λ(n, r). Indeed, the left-hand-side of (2.4.1) is the intersection ∗  µ AK (n, r) ∩ AK (n, r)λ , which may be identified naturally with (µ AK (n, r))∗ ∩ (AK (n, r)λ )∗ . The latter is isomorphic to SK (n, r)1µ ∩ 1λ SK (n, r).

3. Hecke algebras associated to permutation modules We will now study the algebras SK (λ) := EndΣr (M λ ) and the related bimodules µ λ λ (SK )µ := HomΣr (M , M ), for λ, µ ∈ Λ(n, r). 3.1. Tensor space E⊗r has a decomposition  E⊗r = Mλ λ∈Λ(n,r)

(3.1.1)

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

509

as right KΣr -modules. Here M λ is the transitive permutation module with basis all ei = ei1 ⊗ · · · ⊗ eir (i ∈ I (n, r)) such that wt(i) = λ. Note that M λ is isomorphic to the r Σr -module 1Σ Σλ induced from the Young subgroup Σλ = Σλ1 × · · · × Σλn corresponding to λ, so the algebra SK (λ) is a type of Iwahori–Hecke algebra; see also [17, 6.1, Remark]. The idempotents 1λ of SK (n, r) as in (2.2.3) are precisely the projections corresponding to the direct sum decomposition (3.1.1); see [17, 3.2]. Thus we have an isomorphism of SK (λ)-SK (µ) bimodules   1λ SK (n, r)1µ  HomΣr M µ , M λ

(3.1.2)

and in particular an algebra isomorphism   1λ SK (n, r)1λ  EndΣr M λ .

(3.1.3)

Hence we may identify SK (λ) with 1λ SK (n, r)1λ and, more generally, λ (SK )µ with 1λ SK (n, r)1µ . Moreover, whenever n
r, we have an isomorphism 1ω SK (n, r)1ω  KΣr ,

(3.1.4)

where ω ∈ Λ(n, r) is the special weight ω = (1, . . . , 1, 0, . . . , 0) (r 1’s),

(3.1.5)

since M ω is the regular representation of KΣr . (See [17, (6.1d)] for an explicit isomorphism.) The following lemma shows that one need only consider SK (λ) and λ (SK )µ in case λ, µ are dominant, that is λ1
λ2
· · ·
λn , since such λ label the Σn -orbits of Λ(n, r). Lemma 3.2. Let λ, µ ∈ Λ(n, r). (a) For any w ∈ Σn , there is an algebra isomorphism SK (λ)  SK (wλ). (b) For any w, w ∈ Σn , there is an isomorphism λ (SK )µ  wλ (SK )w µ of SK (λ)-SK (µ) bimodules. Proof. It is enough to show that M λ  M wλ for any w ∈ Σr . But this is clear from the definition of M λ (see 3.1). 2 From the isomorphism (3.1.2) and Schur’s product rule [17, (2.3c)] we immediately obtain the following result, which provides a basis for λ (SK )µ . Proposition 3.3. Let λ, µ ∈ Λ(n, r). A basis for 1λ SK (n, r)1µ is given by the set of all ξi,j such that wt(i) = λ, wt(j) = µ. In particular, the set of all ξi,j (i, j ∈ I (n, r)) satisfying wt(i) = λ = wt(j) is a basis for SK (λ).

510

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

In [18], J.A. Green introduced the codeterminant basis of SK (n, r). There is a similar basis for SK (λ). Let ν ∈ Λ(n, r) be a composition. Given a word i ∈ I (n, r) let Tiν denote the ν-tableau obtained by writing the components of i in left to right order across successive rows in the Young diagram of shape ν. The weight of a tableau Tiν is simply wt(i). The multi-index (ν) is the word consisting of ν1 1’s, followed by ν2 2’s, and so forth. The following is immediate from the main result of [18]. Proposition 3.4. For λ, µ ∈ Λ(n, r), there is a basis for 1λ SK (n, r)1µ consisting of the ν codeterminants of the form Yi,j = ξi, (ν)ξ (ν),j , such that ν ∈ Λ(n, r) is dominant and Tiν , ν Tj are semistandard tableaux of shape ν and weight λ, µ, respectively. This shows again (see [22, 13.19]) that the dimension of SK (λ) is the number of pairs of semistandard tableaux (of some dominant shape ν) of weight λ. Our next result says that HomGLn (S µ E, S λ E) is isomorphic to HomΣr (M λ , M µ ) and, in particular, EndSK (n,r) (S λ E) is isomorphic to EndΣr (M λ ). This was proved in case n
r in [6, 2.4]; the present version holds without that assumption. Proposition 3.5. Let n and r be arbitrary positive integers, E = K n . Regard S λ E as a left Mn -module (and hence a left GLn -module by restriction). (a) For λ ∈ Λ(n, r) there is an isomorphism of algebras   EndGLn S λ E  SK (λ). (b) For λ, µ ∈ Λ(n, r) there is an isomorphism of SK (λ)-SK (µ) bimodules   HomGLn S µ E, S λ E  1λ S(n, r)1µ . Proof. Obviously restriction induces an isomorphism     HomGLn S µ E, S λ E  HomMn S µ E, S λ E . By Lemma 1.3 we have     Mn n HomMn S µ E, S λ E  HomMn indM Dn λ K, indDn µ K . By Frobenius reciprocity and the second part of (1.1.3) we have   ∗    Mn n HomMn indM Dn λ K, indDn µ K  HomDn K[Mn ]λ , µ K  µ K[Mn ]λ and by Lemma 2.3(a) and the isomorphism (2.4.1) this is in turn isomorphic with 1λ SK (n, r)1µ . Taking λ = µ proves part (a). Since HomGLn (S µ E, S λ E) is naturally a bimodule with EndGLn (S λ E) acting on the left and EndGLn (S µ E) acting on the right, the isomorphism of part (a) defines an SK (λ)-SK (µ) bimodule structure on HomGLn (S µ E, S λ E), and part (b) follows. 2

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

511

Remarks 3.6. (a) When n
r the Schur functor takes S λ E to M λ (where M λ is viewed as left Σr -module). Hence it induces a bimodule map     HomSK (n,r) S µ E, S λ E → HomΣr M µ , M λ (an algebra map in case λ = µ). In [6, 2.4] it is proved that the map is an isomorphism, by first arguing that the map is injective (since the modules are tilting) and then comparing dimensions. The preceding proposition obtains this isomorphism anew, without using the Schur functor, and with no restriction on n, r. (b) If n
r we may take λ = µ = ω in part (a) of Proposition 3.5, obtaining an isomorphism EndGLn (E⊗r )  SK (ω). In other words, EndGLn (E⊗r )  KΣr (see (3.1.4)). (c) Since the natural (left) action of GLn on E⊗r commutes with the (right) action of Σr by place-permutations, the image of the anti-representation   σ : KΣr → EndK E⊗r is contained in EndGLn (E⊗r ). One part of Schur–Weyl duality states that this containment is equality. In other words, the induced map     σ : KΣr → EndGLn E⊗r  EndSK (n,r) E⊗r is surjective. For general K this seems to go back to [3], but in case n
r it appeared earlier in [2]. A different proof (with no restriction on n, r) was given recently in [24]. We will now show how this result follows from results of [15]. Assume first that n
r. Then it is clear that E⊗r is faithful as a module for the symmetric group; that is, σ is injective. By remark (b) it follows that σ is then an isomorphism. For the general case, take some N
n, r. Let E = K N be the natural GLN (K)-module, and view E ⊂ E , so that we have a projection e ∈ SK (N, r) with e (E ⊗r ) = E⊗r , as in [15, 3.9]. Moreover, we have an isomorphism e SK (N, r)e  SK (n, r). Then [15, 1.7(i)] applies, and the idempotent functor V → e V induces a surjective algebra map  ⊗r    EndSK (N,r) E → EndSK (n,r) E⊗r . That is, any SK (n, r)-endomorphism h of E⊗r is the restriction of some  ⊗r  h ∈ EndSK (N,r) E . By (b) there is some x ∈ KΣr such that for all m ∈ E ⊗r we have (m)h = mx. If m ∈ E⊗r then (m)h = (m)h = mx ∈ E⊗r . In other words, h is the map m → mx. This shows that σ is surjective. (d) In fact, the map σ : KΣr → EndK (E⊗r ) considered in remark (c) is obtained by change of base ring from the natural map ZΣr → EndZ (E⊗r Z ), where EZ is an appropriate

512

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

integral form of EQ (the Z-span of the standard basis), and its image is contained in ⊗r EndGLn (Z) (E⊗r Z ) = EndSZ (n,r) (EZ ). Since the induced map σ is surjective for every infinite field K, it follows that it is surjective when taken over Z. 4. Simple SK (λ)-modules There is a connection between Kostka duality and the problem of parametrizing the simple SK (λ)-modules. 4.1. Let λ ∈ Λ(n, r) be given. Consider now the ‘Schur functor’ Fλ from left SK (n, r)modules to left SK (λ)-modules defined by V → 1λ V . Let L(µ) be the simple SK (n, r)module corresponding to a dominant weight µ ∈ Λ(n, r). By [17, (6.2b)], the collection of all nonzero Fλ L(µ) forms a complete set of simple SK (λ)-modules. 4.2. For a dominant µ ∈ Λ(n, r), let J (µ) denote the injective envelope of L(µ) in the category of rational Mn -modules. This is the contravariant dual of the projective cover of L(µ) in the category of rational Mn -modules (or the category of SK (n, r)-modules). By the n remarks following (1.1.3), the generalized symmetric power S λ E  indM Dn Kλ is injective λ as an S(n, r)-module, for any λ ∈ Λ(n, r). Write (S E : J (µ)) for the multiplicity of J (µ) in a Krull–Schmidt decomposition of S λ E. By Frobenius reciprocity one has an equality  λ  S E : J (µ) = dimK 1λ L(µ). (4.2.1) (For more details see [12] or [7, (3.4)].) The equality is known as Kostka duality; the nonnegative integer in the equality is the Kostka number, denoted by Kµλ . Note that Kµλ may be equivalently defined to be the multiplicity of a Young module Y µ in a Krull– Schmidt decomposition of M λ , see [7, (3.5), (3.6)]. Proposition 4.3. Let λ ∈ Λ(n, r) be fixed. Up to isomorphism, the simple SK (λ)-modules are the λ L(µ) = 1λ L(µ) for which Kµλ = 0. Proof. By [17, (6.2b)] the set of isomorphism classes of simple SK (λ)-modules are precisely the nonzero modules of the form Fλ L, for L a simple SK (n, r)-module. The result now follows by Kostka duality. 2 Remark 4.4. Donkin [7, p. 55, Remark] gives a more precise necessary and sufficient condition on λ, µ for Kµλ = 0, obtained by combining Steinberg’s tensor product theorem with Suprunenko’s theorem [31].

5. PBW basis In this section, we work with the Lie algebra gln and its enveloping algebra U over the rational field Q. We also consider the algebra of distributions (the hyperalgebra) UZ of the affine algebraic Z-group scheme GLn corresponding to the algebra Z[cij ; (det(cij ))−1 ].

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

513

5.1. Let eij be the n×n matrix whose unique nonzero entry is a 1 in the (i, j )th position. Set Hi := eii and fij := ej i . The set {eij | i < j } ∪ {Hi } ∪ {fij | i < j }

(5.1.1)

is a Chevalley basis for gln . We regard these as elements of the universal enveloping algebra U = U(gln ). Note that U is generated by the elements ei = ei,i+1 ,

fi = ei+1,i

(1  i  n − 1),

Hi

(1  i  n).

(5.1.2)

For an element X and an integer a
0, set  X(X − 1) · · · (X − a + 1) X Xa (a) X := , := . a a! a! (a)

The hyperalgebra UZ of the Z-group GLn is the Z-subalgebra of U generated by all fi , Hi  (c) b , ei (a, b, c
0); see [23, II, Chapter 1]. The set of all products of the form

(aij )

fij

i
i



Hi (c ) eij ij bi

(5.1.3)

i
for nonnegative integers aij , bi , forms a Z-basis of UZ , where the products among the f ’s (respectively, e’s) are taken in some arbitrary, but fixed, order. With similar conventions on the order of products, the set

(cij ) Hi (aij ) eij fij (5.1.4) bi i
i

i
is another Z-basis of UZ . 5.2. By differentiating the representation GLn → EndQ (E⊗r ) one obtains a representation gln → EndQ (E⊗r ); this extends uniquely to a representation   πn,r U −−→ EndQ E⊗r

(5.2.1)

and S(n, r) (over Q) is the image of this representation. Moreover, SZ (n, r) is the image of UZ under the same map. In fact, one can consider the UZ -invariant lattice EZ (the Z-span of the standard basis (ei ) on E) and check that the map πn,r takes UZ into EndZ (E⊗r Z ); then SZ (n, r) is the image πn,r (UZ ) [2]. One has isomorphisms SK (n, d)  K ⊗Z SZ (n, r),

UK  K ⊗Z UZ

(5.2.2)

for a field K, where UK is the hyperalgebra of the K-group (GLn )K corresponding to the algebra K[cij ; (det(cij ))−1 ]. Thus SK (n, r) is a homomorphic image of UK . Denote the   images in S(n, r) of the elements fij(a) , Hbi , eij(c) by the same symbols.

514

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

Lemma  Hi  5.3. The element 1λ in S(n, r) coincides with the element given by the product i λi , for any λ ∈ Λ(n, r). Proof. The element 1λ was defined in (2.2.3). It is known (see [17, 3.2]) that, if M is ∈ Λ(n, any S(n, r)-module, then 1λ M coincides with the λ-weight space of M (for λ  i  r)), considered as left GLn -module. Similarly, one can show that the element 1λ := i H λi acts as zero on all weight space except the λ one, and acts as 1 there. This proves that 1λ and 1λ act the same on all modules M. To finish, we need the existence of a faithful module, since on such a module the difference 1λ − 1λ acts as zero, hence must equal zero in the algebra. Since E⊗r is faithful, as S(n, r)-module, the proof is complete. 2 5.4. [11, Theorem 2.3] can be reformulated as follows. Given an n×n matrix A = (aij ),  + ), where λ+ = a + let λ+ (A) := (λ+ , . . . , λ (a jj n ij (aij + aj i ) for each j . Let Θ(n, r) be the set of n × n matrices over N whose entries sum to r. Then by [11, Theorem 2.3] the set of products of the form 

(aji )

fij



  (a ) A ∈ Θ(n, r) eij ij 1λ− (A)

i
(5.4.1)

i
is a Z-basis for SZ (n, r). Similarly, the set 

(a )

eij ij



  (a ) 1λ+ (A) A ∈ Θ(n, r) fij ji

i
(5.4.2)

i
is another basis for SZ (n, r). As usual, one has to take products of f ’s (respectively, e’s) with respect to an arbitrary, but fixed, order. One can easily see that any element of the form (5.4.2) has content (in the sense defined in [11, p. 1911]) not exceeding λ+ (A); conversely, given a monomial of the form eA 1µ fC with content not exceeding µ, one can find a matrix A ∈ Θ(n, r) which defines that monomial. This proves that the first basis given in [11, (2.7)] coincides with the basis described in (5.4.2) above. One can argue similarly that the second basis in [11, (2.7)] coincides with the basis (5.4.1).1 By commutation relations given in [11, Proposition 4.5] one can rewrite a given basis element of the form (5.4.1) in the form

1row(A)



i
(aji ) (aij ) eij i
fij

1col(A)

  A ∈ Θ(n, r) ;

(5.4.3)

1 There is an error in the definition of the second basis in [11, (2.7) and (3.9)]: for a correct description of the second basis, one needs a definition of content in which max is replaced by min.

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

515

similarly one can rewrite a given basis element of the form (5.4.2) in the form 1row(A)



(a )

eij ij

i
(aji )

fij

1col(A)

  A ∈ Θ(n, r)

(5.4.4)

i
where row(A) (respectively, col(A)) is the vector of row (respectively, column) sums in the matrix A. Identify S(λ) with 1λ S(n, r)1λ and write SZ (λ) for 1λ SZ (n, r)1λ ; similarly write λ (SZ )µ for 1λ SZ (n, r)1µ . Then λ (SK )µ  K ⊗Z λ (SZ )µ . The following result provides a basis for SK (λ), over any field K. Proposition 5.5. For λ, µ ∈ Λ(n, r) the bimodule λ (SZ )µ has a Z-basis consisting of all elements 

(a ) (aij ) 1λ fij ji eij (5.5.1) 1µ (A ∈ λ Θµ ) i
i
and another such basis consisting of all 1λ



(a ) (aji ) eij ij fij i
1µ

(A ∈ λ Θµ )

(5.5.2)

where λ Θµ is the set of all n × n matrices over N with row and column sums equal to λ and µ respectively, and the products of f ’s (respectively, e’s) is taken with respect to some fixed, but arbitrary, order. Proof. This follows immediately from (5.4.3), (5.4.4) above. 2

6. Lusztig’s modified form of U(gln ) We work over Q in this section. We consider the modified form of U = U(gln ). 6.1. Recall that U is the associative algebra with 1 given by generators ei , fi (1  i  n − 1), and Hi (1  i  n) subject to the usual relations given by the Lie algebra structure (see, e.g., [11, relations (R1)–(R5)]): Hi Hj = Hj Hi ,

ei fj − fj ei = δij (Hj − Hj +1 ),

Hi fj − fj Hi = −εi,j fj , Hi ej − ej Hi = εi,j ej ,   ei2 ej − 2ei ej ei + ej ei2 = 0 |i − j | = 1 , ei ej = ej ei (otherwise),   fi fj = fj fi (otherwise) fi2 fj − 2fi fj fi + fj fi2 = 0 |i − j | = 1 ,

516

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

where εi,i = 1, εi,i−1 = −1, and εi,j = 0 for j = i, i − 1. For λ, µ ∈ Zn we define λ Uµ (as a vector space) to be the quotient: λ Uµ

=U



(Hi − λi )U +

i



U(Hi − µi ) .

(6.1.1)

i

Let πλ,µ : U → λ Uµ be the  canonical projection. For convenience of notation, set λ I =  (H − λ )U and I = i i µ i i U(Hi − µi ). Then λ Uµ = U/(λ I + Iµ ). Set αi = εi − εi+1 for 1  i  n − 1, where {ε1 , . . . , εn } is the canonical basis of Zn . Consider the grading on U defined by putting Hi ∈ U[0],
ei ∈ U[αi ], fi ∈ U[−αi ] subject  ]U[ν  ] ⊂ U[ν  + ν  ]. Then U = to the requirement U[ν ν U[ν], where ν runs over the set  Zαi (the root lattice). For ν an element of the root lattice, let λ Uµ [ν] be the image of U[ν] under πλ,µ . We have λ Uµ

=



λ Uµ [ν].

(6.1.2)

ν

One may check from the defining relations for U given above that ej (λ I ) ⊂ λ+αj I,

fj (λ I ) ⊂ λ−αj I,

Hj (λ I ) ⊂ λ I,

(Iµ )ej ⊂ Iµ−αj ,

(Iµ )fj ⊂ Iµ+αj ,

(Iµ )Hj ⊂ Iµ .

(6.1.3)

From this it follows by induction that U[ν] · λI ⊂ λ+ν I and Iµ ·U[ν] ⊂ Iµ−ν for any element ν in the root lattice. The following observation is important for what follows. Lemma 6.2. λ Uµ [ν] is zero unless λ − µ = ν. Proof. Let u ∈ U[ν]. We must show that u ∈ λ I + Iµ unless λ − µ = ν. We have (Hi − λi )u ∈ λ I and u(Hi − µi ) ∈ Iµ . But from the defining relations for U we have by induction the equality (Hi − λi )u = u(Hi − λi + νi ) for all i. Now for any integers a = b we have 1=

1 1 (Hi − a) + (Hi − b). b−a a−b

Taking a = λi − νi , b = µi we see that unless λi − νi = µi for each i, we have u = u · 1 ∈ + Iµ . The lemma is proved. 2

λI

6.3. The above
grading on U allows one to define a natural associative Q-algebra structure on U˙ = λ,µ (λ Uµ ), inherited from that of U. For any λ, µ, λ , µ ∈ Zn and any t ∈ U[λ − µ], s ∈ U[λ − µ ], the product πλ,µ (t)πλ ,µ (s) is defined to be equal to πλ,µ (ts) if µ = λ and is zero otherwise. From (6.1.3) one verifies that the multiplication is welldefined.
˙ For ν an element of the root lattice, set U[ν] = λ,µ λ Uµ [ν].

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

517

The elements 1λ = πλ,λ (1) satisfy the relation 1λ 1µ = δλ,µ 1λ , i.e., {1λ } is a family of ˙ The algebra U˙ does not have 1 (for n
2). We have the orthogonal idempotents in U. equalities λ Uµ

for λ, µ ∈ Zn , ν ∈ U˙ =



˙ µ, = 1λ U1

˙ λ Uµ [ν] = 1λ U[ν]1 µ

(6.3.1)

Zαi . Thus we have the decompositions



˙ U[ν],

U˙ =

ν



˙ µ, 1λ U1

U˙ =

λ,µ



˙ 1λ U[ν]1 µ

(6.3.2)

λ,µ,ν

where λ, µ vary over Zn and ν varies over the root lattice. ˙ defined by the requirement 6.4. There is a natural U-bimodule structure on U,   tπλ,µ (s)t  = πλ+ν,µ−ν  tst 

(6.4.1)

for all t ∈ U[ν], t  ∈ U[ν  ], λ, µ ∈ Zn . Moreover, the following identities hold in the algeb˙ ra U: ei 1λ = 1λ+αi ei ,

fi 1λ = 1λ−αi fi ,

(6.4.2)

(ei fj − fj ei )1λ = δij (λi − λi+1 )1λ

(6.4.3)

for all λ ∈ Zn , all i, j . One can show that the algebra U˙ is generated by all elements of the form ei 1 λ ,

fi 1λ



 1  i  n − 1, λ ∈ Zn .

(6.4.4)

The algebra U˙ inherits a “comultiplication” from the comultiplication on U (follow [26, 23.1.5]) but we shall not need it here. ˙ 6.5. Following Lusztig [26, 23.1.4] we say that a U-module M is unital if: n (a) for all m ∈ M one has 1 λ m = 0 for all but finitely many λ ∈ Z , n (b) for any m ∈ M one has λ 1λ m = m (sum over all λ ∈ Z ).

˙ If M is a unital U-module, then one may regard it as a U-module
with weight space decomposition, as follows. The weight space decomposition M = (λ M) is given by setting λ M = 1λ M and the action of u ∈ U is defined by u · m = (u1λ )m for any m ∈ λ M, where u1λ is regarded as an element of U˙ as in 6.4. ˙ From the above one sees that the category of unital U-modules is the same as the category of U-modules admitting weight space decompositions.

518

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

6.6. Let U˙ Z be the subalgebra of U˙ generated by all (c)

ei 1 λ ,



(a)

fi 1λ

 1  i  n − 1, λ ∈ Zn , a, c
0 .

(6.6.1)

− − Let B + be any Z-basis of U+ Z and B any Z-basis of UZ . By the analogue of [26, 23.2.1] ˙ we have a Z-basis for UZ consisting of all elements of the form

b+ 1λ b−



b + ∈ B + , b− ∈ B − , λ ∈ Zn



(6.6.2)

and another such basis consisting of the elements of the form  +  b ∈ B + , b− ∈ B − , λ ∈ Zn .

b− 1λ b+

(6.6.3)

It follows that there is a Z-basis for U˙ Z consisting of all elements of the form 

(aji )

fij

i


  (a )  1λ− (A) A ∈ Θ(n) eij ij

(6.6.4)

i
and another Z-basis consisting of all elements of the form 

(a ) eij ij

i


  (aji )  A ∈ Θ(n) fij 1λ+ (A)

(6.6.5)

i
 where Θ(n) is the set of n × n integral matrices with off-diagonal entries
0, and where as usual products of f ’s (respectively, e’s) are taken with respect to some fixed order. The definition of λ+ (A), λ− (A) here is the same as in 5.4. ˙ 7. The generic algebra U(λ) ˙ λ 7.1. From the definition (see 6.3) of the multiplication in U˙ it is clear that λ Uλ = 1λ U1 ˙ is an algebra and λ Uµ = 1λ U1µ is a bimodule with λ Uλ acting on the left and µ Uµ acting on the right. By definition, U˙ is the direct sum of these bimodules, as λ, µ range over Zn . ˙ ˙ λ ; this is the generic algebra. It is both a quotient of U and a subalgebra Set U(λ) := 1λ U1 ˙ of U. We have the following analogue of Lemma 3.2, which shows that one needs to consider ˙ U(λ) and λ Uµ only in case λ, µ ∈ Zn are dominant. Lemma 7.2. Let λ, µ ∈ Zn . ˙ ˙ (a) For any w ∈ Σn , one has an algebra isomorphism U(λ)  U(wλ). ˙ ˙ (b) For any w, w ∈ Σn , one has an isomorphism λ Uµ  wλ Uw µ of U(λ)U(µ) bimodules.

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

519

Proof. The Weyl group W  Σn acts as permutations on the Lie algebra gln via the rule: weij = ew−1 (i),w−1 (j ) . (Here the eij are the matrix units defined in 5.1.) This induces a corresponding action of Σn on U. Using this action, the statements are easily checked. 2 The following result provides a basis for U˙ K (λ) := U˙ Z (λ) ⊗Z K under specialization to any field K. Proposition 7.3. For any λ, µ ∈ Zn , a Z-basis for the bimodule λ (UZ )µ = 1λ U˙ Z 1µ is given by the set of all elements of the form 1λ



(a ) (aij ) fij ji eij i
1µ



µ A ∈ λΘ



(7.3.1)

and another such basis is given by all elements of the form 1λ



(a ) (aji ) eij ij fij i
1µ



µ A ∈ λΘ



(7.3.2)

 µ is the set of all matrices A ∈ Θ(n) where λ Θ with row and column sums equal to λ and µ respectively. As usual, products of f ’s (respectively, e’s) is taken with respect to some fixed, but arbitrary, order. Proof. This follows from (6.6.4), (6.6.5) and commutation formulas (6.4.2). The argument is similar to the argument for 5.5. 2 Example 7.4. We consider the case n = 2. Let λ = (λ1 , λ2 ) ∈ Z2 . Then the basis of U˙ Z (λ) described in (7.3.1) consists of all elements of the form 1λ f (a) e(a)1λ

(a
0)

(7.4.1)

and the basis described in (7.3.2) consists of all elements of the form 1λ e(a)f (a) 1λ

(a
0).

(7.4.2)

If λ ∈ Λ(2, r), the nonzero images in SZ (λ) of the elements (7.4.1) give the following set of elements 1λ f (a) e(a)1λ

  0  a  min(λ1 , λ2 )

(7.4.3)

which was described in (5.5.1); the nonzero images in SZ (λ) of the elements (7.4.2) give the following set of elements 1λ e(a) f (a) 1λ

  0  a  min(λ1 , λ2 )

(7.4.4)

520

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

which was described in (5.5.2). The sets in (7.4.3), (7.4.4) are bases of SZ (λ). Thus dim S(λ) = 1 + min(λ1 , λ2 ). ˙ Note that from this description it follows that U(λ) (for λ ∈ Z2 ) is generated by the element 1λ f e1λ . It is also generated by the element 1λ ef 1λ . It follows that the algebra ˙ U(λ) is commutative. The same statements apply to S(λ) (for λ ∈ Λ(2, r)). This does not hold for n > 2, in general. ˙ K (λ)-modules 8. Simple U We work over an arbitrary infinite field K in this section. Set U˙ K = K ⊗Z U˙ Z , and similarly for U˙ K (λ), λ (UK )µ . The simple unital U˙ K -modules are the same as the simple rational GLn (K)-modules. This follows, for instance, from the v = 1 analogue of [26, Chapter 31]. (Alternatively, see the discussion in [23, II, 1.20].) 8.1. Let λ ∈ Zn be given. Let Fλ be the functor V → 1λ V from unital U˙ K -modules to left U˙ K (λ)-modules. By [17, 6.2] this is an exact covariant functor mapping simple modules to simple modules or 0. Let L(µ) be the simple U˙ K -module corresponding to a dominant weight µ ∈ Zn . By [17, (6.2b)], the collection of all nonzero Fλ L(µ) forms a complete set of simple U˙ K (λ)-modules. 8.2. We write GLn = GLn (K). For a dominant µ ∈ Zn , let I (µ) denote the injective envelope of L(µ) in the category of rational GLn -modules. As already observed in 1.1, GLn n n indGL Tn Kλ is injective as rational GLn -module, for any λ ∈ Z . Write (indTn Kλ : I (µ)) GLn for the multiplicity of I (µ) in a Krull–Schmidt decomposition of indTn Kλ . By Frobenius reciprocity one has an equality 

 n indGL Tn Kλ : I (µ) = dimK 1λ L(µ)

(8.2.1)

and this shows, in particular, that the multiplicities on the left-hand-side are finite. Let us denote the integer in the equality by Kµλ . The following analogue of 4.3 is now clear. Proposition 8.3. Let λ ∈ Zn be fixed. The isomorphism classes of simple U˙ K (λ)-modules are the λ L(µ) = 1λ L(µ) for which Kµλ = 0. ˙ 9. S(λ) is a quotient of U(λ) ˙ Now we consider connections between Schur algebras and the decomposition of U. Fix positive integers n, r and write λ Sµ = 1λ S(n, r)1µ for λ, µ ∈ Λ(n, r). Let πn,r : U → S(n, r) be the surjective algebra map as in (5.2.1). In this section we use the notation 1λ for two different objects. Since 1λ ∈ S(n, r) is a homomorphic image of the corresponding 1λ ∈ U˙ this should cause no confusion.

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

521

Proposition 9.1. (a) For any λ, µ ∈ Λ(n, r) the linear map ψλ,µ : U → 1λ S(n, r)1µ given by u → 1λ πn,r (u)1µ is surjective. Moreover, ψλ,µ induces a linear surjection λ Uµ = ˙ µ → 1λ S(n, r)1µ . 1λ U1 (b) The restriction to UZ of ψλ,µ surjects onto 1λ SZ (n, r)1µ . ˙ (c) For λ ∈ Λ(n, r), S(λ) is a quotient algebra of U(λ). Similarly, SZ (λ) is a quotient ˙ algebra of UZ (λ). Proof. The surjectivity of ψλ,µ is clear from its definition and the surjectivity of πn,r . By [11, Proposition 4.3(a)], in S(n, r) we have the relation Hi 1µ = 1µ Hi = µi 1µ , for any i and any µ ∈ Zn . (Here, Hi is really the image of Hi in S(n, r).) It follows that Hi − λi acts on the left on 1λ S(n, r)1µ as zero;  similarly, Hi −µi acts as zero on the right. Thus the map ψλ,µ sends any element of i (Hi − λi )U + i U(Hi − µi ) to zero, so ψλ,µ factors through λ Uµ , i.e. there is a commutative diagram

ψλ,µ

U

λ Sµ

(9.1.1) λ Uµ

in which all maps are surjections. This proves part (a). By the remarks in 5.2 the map πn,r maps UZ surjectively onto SZ (n, r), so ψλ,µ maps UZ surjectively onto 1λ SZ (n, r)1µ . This proves (b). The first statement in (c) is the special case µ = λ of part (a) of the proposition (one easily checks that the induced map is an algebra map in this case), and the second statement in (c) follows from (b). The proposition is proved. 2 Remark 9.2. Although U and S(λ) are algebras, the map ψλ,λ : U → S(λ) is not in general an algebra map. For instance, the product fi ei may map to a nonzero element of S(λ) but ei and fi themselves always map to zero.  9.3. For
λ, µ ∈ Λ(n, r) and ν ∈ i Zαi , let λ Sµ [ν] be the image of U[ν] under ψλ,µ . Set S[ν] = λ,µ λ Sµ [ν]. Since (the images of) ei , fi , Hi generate S(n, r), it follows that S[ν] is equivalently determined by the requirements ei ∈ S[αi ], fi ∈ S[−αi ], Hi ∈ S[0], and S[ν]S[ν  ] ⊂ S[ν + ν  ]. Obviously

S(n, r) =

 ν

S[ν] =

 λ,µ,ν

λ Sµ [ν],

(9.3.1)

522

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

where ν varies over the root lattice and λ, µ over Λ(n, r). By restriction in diagram (9.1.1) we obtain for each ν a commutative diagram λ Sµ [ν]

U[ν]

(9.3.2) λ Uµ [ν]

in which all maps are surjections. We note that from Lemma 6.2 it follows that λ Sµ [ν] = 0

whenever λ − µ = ν.

(9.3.3)

We now obtain the following. Proposition 9.4. Let λ ∈ Λ(n, r). (a) The restriction of ψλ,λ to the subalgebra U[0] of U is an algebra map which factors through λ Uλ . In other words, there is a commutative diagram U[0]

λ Sλ

λ Uλ

of algebra surjections. Thus the algebra S(λ) = λ Sλ is a homomorphic image both of ˙ U[0] and U(λ) = λ Uλ . (b) The algebra SZ (λ) is a homomorphic image of UZ [0] and of U˙ Z (λ). (c) In particular, for n
r, the algebra ZΣr is a quotient of the algebra UZ [0] and of the algebra U˙ Z (ω). Proof.
Take µ = λ and ν = 0 in (9.3.2).
By Lemma 6.2 and (9.3.3) we have equalities U = U [ν] = U [0] and S = diagram λ λ λ λ λ λ νλ λ ν λ Sλ [ν] = λ Sλ [0], so the commutative
in (a) results. These maps are algebra maps, since πn,r (U[0]) = S[0] = λ λ Sλ and hence 1λ πn,r (u) = πn,r (u)1λ , for any u ∈ U[0]. This proves (a). Part (b) follows from part (a) and Proposition 9.1(b). Part (c) is immediate from part (b) and the identification ZΣr  SZ (ω) (see Remark 3.6(d)). 2 The proof of the preceding result gives the following. ˙ Corollary 9.5. The algebras U[0] and S[0] have decompositions ˙ U[0] =

 λ∈Zn

˙ U(λ),

S[0] =

 λ∈Λ(n,r)

S(λ).

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

523

Proof. Since λ Uµ
[0] = 0 and λ Sµ [0] = 0
unless λ = µ, the result follows from the ˙ ˙ equalities U[0] = λ,µ λ Uµ [0] and S[0] = λ,µ λ Sµ [0] and the identifications U(λ) = λ Uλ = λ Uλ [0], S(λ) = λ Sλ = λ Sλ [0] obtained in the proof of part (a) of the preceding proposition. 2 ˙ Thus we see that the balanced part U[0] (respectively, S[0]) of the algebra U˙ ˙ (respectively, S(n, r)) considers the representations of all the generic algebras U(λ) (respectively, all the Hecke algebras S(λ)) jointly.

10. The modified form of U(sln ) 10.1. By the PBW theorem, U(sln ) may be identified with the subalgebra of U = U(gln ) generated by all ei , fi , hi := Hi − Hi+1 (1  i  n − 1). Given a weight λ ∈ Zn (for gln ) we obtain a corresponding weight λ˜ ∈ Zn−1 (for sln ) as follows: λ → λ˜ := (λ1 − λ2 , λ2 − λ3 , . . . , λn−1 − λn ). ˙ n ) is nearly the same as the definition of U(gl ˙ n ). For λ˜ , µ˜ ∈ Zn−1 one The definition of U(sl defines λ˜ U(sln )µ˜ (as a vector space) to be the quotient space: U(sln )

     ∨   U(sln ) hi − µα ˜ i . hi − λ˜ αi∨ U(sln ) + i

(10.1.1)

i


˙ n ) = ˜ ( ˜ U(sln )µ˜ ). The rest of the construction is exactly the same as for Set U(sl λ,µ˜ λ ˙ n ). The only essential difference between U(gl ˙ n ) and U(sl ˙ n ) is that the former has U(gl more idempotents than the latter. 10.2. The restriction to U(sln ) of the map πn,r is still a surjection onto SQ (n, r). This follows from the decomposition gln = sln ⊕ QI ; the image of I in U(gln ) acts as scalars. Hence the Schur algebra SQ (n, r) is a homomorphic image of U(sln ). By 5.2, this restriction is compatible with integral forms; i.e., the restriction of πn,r to UZ (sln ) surjects onto SZ (n, r). Proposition 10.3. For λ, µ ∈ Zn , the natural map λ˜ UZ (sln )µ˜ → λ UZ (gln )µ is an isomorphism of bimodules. In case µ = λ it is an isomorphism of algebras. Proof. One can adapt the argument for [26, 23.2.5] to prove the map is a Z-linear isomorphism. In fact, the sln -analogue of Proposition 7.3 holds (with a similar proof): there are Z-bases of λ˜ UZ (sln )µ˜ of the form (7.3.1) and (7.3.2), with 1λ , 1µ there replaced by 1λ˜ , 1µ˜ . The bijection is now clear. The last claim follows from the compatibility of the ˙ n ), U(sl ˙ n ). 2 isomorphism with the product in U(gl

524

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

Corollary 10.4. Let λ ∈ Zn . With U˙ Z (λ) = λ UZ (gln )λ we have an algebra isomorphism U˙ Z (λ)  U˙ Z (λ + k(1n )), for any k ∈ Z. Proof. This is immediate from the proposition since with λ = λ + k(1n ) we have λ˜ = λ˜ , so both algebras are isomorphic with λ˜ U˙ Z (sln ))λ˜ . 2 Remark 10.5. Let λ˜ ∈ Zn−1 be given. The preceding results show that all the algebras ˜ are quotients of the single generic algebra SZ (λ), for any λ ∈ Nn belonging to the fiber of λ, ˙ n)˜ . ˜ := ˜ U(sl U˙ Z (λ) λ λ ˙ λ), ˜ S(λ) 11. Cellularity of U( Lusztig [25] showed that the positive part U+ of a quantized enveloping algebra U has ˙ We show that the a canonical basis. In [26, Part IV] the canonical basis is extended to U. ˙ ˙ ˜ canonical basis on U induces compatible canonical bases on U(λ), and S(λ). We apply this ˙ λ) ˜ and S(λ) inherit canonical bases from the canonical basis information to show that U( ˙ λ˜ ) and ˙ on U(sln ), and that these bases are cellular bases. In particular, this shows that U( S(λ) are cellular algebras. (One can see the cellularity of S(λ) in other ways; see, e.g., [4, Theorem 6.6], [27, Chapter 4, Exercise 13].) 11.1. We recall from [16] the definition of cellular algebra. Let A be an associative algebra over a ring R (commutative with 1). We do not insist that A has 1, nor do we insist that A be finite-dimensional. A cell datum for A is a quadruple (Λ, M, C, ι) where: (a) Λ (the set of weights) is partially ordered by
, M is a function from Λ to the class of finite sets, and C is an injective function C:

  M(λ) × M(λ) → A λ∈Λ

λ with image an R-basis of A. If λ ∈ Λ and S, T ∈ M(λ) then one writes CS,T for C(S, T ). λ ) = Cλ . (b) The map ι is an R-linear involutory antiautomorphism of A such that ι(CS,T T ,S (c) If λ ∈ Λ and S, T ∈ M(λ) then for all a ∈ A one has λ aCS,T ≡



  ra S  , S CSλ ,T

mod A(> λ),

S  ∈M(λ)

where ra (S  , S) is independent of T and A(> λ) is the R-submodule of A generated µ by the set of all CS  ,T  such that µ > λ, S  , T  ∈ M(µ). λ } Any algebra A that possesses a cell datum is said to be cellular and the basis {CS,T inherent in its cell datum is its cellular basis.

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

525

Remark 11.2. For our applications, it is convenient to use the order on Λ opposite to the order in the usual definition, so we have reversed the usual ordering of weights in the definition above. 11.3. Following R.M. Green [20] we say that a cell datum (Λ, M, C, ι) is of profinite type if Λ is infinite and if for each λ ∈ Λ, the set {µ ∈ Λ: µ  λ} is finite. The following useful lemma follows immediately from the definitions. Lemma 11.4. (a) Let (Λ, M, C, ι) be a cell datum for A. Let e ∈ A be an idempotent fixed by the involution ι, and suppose that eC(S, T )e is either zero or C(S, T ) for any S, T . Then λ e = 0, some eAe is cellular, with cell datum (Λ, M, C, ι), where Λ = {λ ∈ Λ: eCS,T λ e = 0} for any λ ∈ Λ; C is defined by S, T ∈ M(λ)}, M(λ) = {S, T ∈ M(λ): eCS,T λ λ C S,T = eCS,T e whenever λ ∈ Λ, S, T ∈ M(λ), and ι is the restriction of ι to eAe. (b) If the original cell datum (Λ, M, C, ι) is of profinite type, then (Λ, M, C, ι) is also of profinite type, provided Λ is infinite. 11.5. We consider U = U(sln ), the quantized enveloping algebra corresponding to sln . Lusztig [26] shows that the canonical basis on U+ , the plus part of U, induces a canonical ˙ Moreover, in [26, Chapter 29] Lusztig shows that the basis of the modified form U. ˙ canonical basis of U is a cellular basis. This is spelled out in greater detail in [10, 2.5]. Note that this cellular basis has profinite type. In [25] it was pointed out that the canonical basis on U+ (which is an A-basis for U+ A) + corresponds under specialization to a Z-basis of the plus part U+ 1 = UZ of UZ , the integral ˙ consisting of elements of the form form of U(sln ). By [26, 23.2], one has an A-basis of U b + 1λ b− .

(11.5.1)

Similarly one has another A-basis consisting of elements of the form b − 1λ b+ .

(11.5.2)

In both sets of elements above, b+ (respectively, b− ) vary independently over any A-basis − ˙ ˙ of U+ A (respectively, UA ). It follows that specializing v to 1 takes UA to UZ . Moreover, ˙ A of the form (11.5.1) or (11.5.2) will correspond to a Z-basis of U˙ Z . In any A-basis for U ˙ A corresponds under specialization to a Z-basis of U˙ Z ; particular, the canonical basis on U ˙ It is a cellular basis of U˙ Z , of profinite type. we call this the canonical basis of U. Lemma 11.6. With U˙ Z = U˙ Z (sln ), there is a natural quotient map U˙ Z → SZ (n, r), and the nonzero elements in the image of the canonical basis under the map is a Z-basis of SZ (n, r). (We call it the canonical basis of SZ (n, r).) Proof. By [5,6], SZ (n, r) is a generalized Schur algebra, defined by the saturated set π , the set of dominant weights of Zn−1 occurring in the weight space decomposition of E⊗r Z .

526

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

Since SZ (n, r)  U˙ Z /U˙ Z [P ] (see, e.g., [10]), where P is the complement of π in the set of dominant weights, the claim follows by the analogue of [26, 29.2]. 2 Remark 11.7. We have chosen, for simplicity, to obtain the canonical basis on S(n, r) by ˙ n ). In [13] this is approached (for the q-Schur descent from the canonical basis of U(sl algebra) the other way around, by building up from the Kazhdan–Lusztig basis for the Hecke algebra of type A. See [1] for yet another approach. ˙ n ), and by B˙ r the corresponding 11.8. Let us denote by B˙ the canonical basis of U˙ = U(sl ˜ µ˜ ∈ Zn−1 set canonical basis of S(n, r). For λ,     ˙ 1 ˜ b1µ˜ = 0 in U˙ B˙ λ˜ , µ˜ = 1λ˜ b1µ˜ | b ∈ B, λ and for λ, µ ∈ Λ(n, r) set   B˙ r (λ, µ) = 1λ b1µ | b ∈ B˙ r , 1λ b1µ = 0 in S(n, r) . with the system of idempotents; that is, B˙ =   The canonical basis is compatible ˙ ˜ ˜ is a Z-basis for ˙ ˜ ˙ ˙ = B( λ, µ) ˜ and B n−1 r ˜ µ∈Z λ,µ∈Λ(n,r) Br (λ, µ). Obviously B(λ, µ) λ, ˜ ˙ ˙ 1λ˜ UZ 1µ˜ and Br (λ, µ) is a Z-basis for 1λ SZ (n, r)1µ . ˙ λ˜ ) := B( ˙ λ˜ , λ˜ ) and B˙ r (λ) := B˙ r (λ, λ). Write B( Theorem 11.9. ˙ λ) ˜ is a cellular basis for U˙ Z (λ), ˜ of procelluar type. (a) For λ˜ ∈ Zn−1 , B( (b) For λ ∈ Λ(n, r), B˙ r (λ) is a cellular basis for SZ (λ). Proof. This follows immediately from Lemmas 11.4 and 11.6, since the idempotent 1λ is fixed by the involution, and 1λ b1λ is either 0 or b for any canonical basis element b. 2

12. Completions Ideas considered here are related to [1,14,20,21]. We work over an arbitrary infinite field K. 12.1. Let det = det(cij ) ∈ AK (n, r), as in 1.2. Since this element is group-like, the map defined by f → f · det : AK (n, r) → AK (n, r + n)

(12.1.1)

is a coalgebra morphism, and is clearly injective. By taking compositions of these maps we obtain a direct system of injective coalgebra morphisms ϕji

AK (n, i) −→ AK (n, j )

(12.1.2)

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

527

defined for any i, j ∈ N for which there exists k ∈ N with j − i = kn, and where for i = j the map ϕji is identity. The indexing set for the direct system is N, partially ordered by ij

⇔

there exists k ∈ N with j − i = kn.

A (n, r) be the limit (n is fixed) of the direct system (12.1.2). The partial Let O = lim −→ K order on N is the disjoint union of the usual total order imposed
separately on each congruence class modulo n, so we have a decomposition O = 0
t
j

i SK (n, j ) SK (n, i) ←−−

(i  j ).

(12.2.1)

Here ψi = (ϕji )∗ ; we identify SK (n, r) with AK (n, r)∗ . The inverse limit of this inverse system, which we shall denote by  UK (sln ), is an algebra naturally isomorphic with K[SLn ]∗ . Moreover, we have an isomorphism    (12.2.2) UK (sln )  EndSLn K[SLn ] , j

in light of [23, I, 3.7]. It is clear that all of this is compatible with change of base ring. In fact, the entire A (n, r) and  UZ (sln )  construction may be carried out over Z. So we have Z[SLn ]  lim −→ Z EndSLn (Z[SLn ]).   The algebra UK (sln ), as an inverse limit, may be taken as the set of all (ur ) ∈ S (n, r) satisfying the condition r∈N K j

(12.2.3) ψi (uj ) = ui whenever i  j in N.  There is a natural map UK (gln ) → r∈N SK (n, r) defined by u → (πn,r (u)). However,  Hi the image of this map is not contained in  UK (sln ) since the image of the element

528

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

 does not satisfy condition (12.2.3). In fact, from [11] we know that πn,r ( Hi ) = r, j for any r ∈ N, and the quotient map ψi , being a linear map, surely cannot map j to i unless i = j . This shows the inverse system is not compatible with the quotient maps πn,r : UK (gln ) → SK (n, r). However, restricting to UK (sln ) we find that such compatibility does hold. Proposition 12.3.  (a) The image of the natural algebra map UK (sln ) → r∈N SK (n, r), defined by u → (πn,r (u)), is contained in  UK (sln ). This map is injective, so UK (sln ) is isomorphic with a subalgebra of  UK (sln ). (b) The algebra  UK (sln ) may be identified with the algebra of all formal (possibly infinite) linear combinations of the canonical basis B˙ (see 11.8). The unit element in  UK (sln ) is the sum of all the idempotents 1λ˜ (λ˜ ∈ Zn−1 ). Proof. (a) Follow the argument of Theorem 6.4.12 in R.M. Green’s dissertation [19]. (b) We use the notation of 11.8. The map in (a) factors through U˙ K (sln ) (using the ˙ K (sln ) → SK (n, r) as in 11.6). We want to extend this to the completion with map π˙ n,r : U respect to the canonical basis.  Consider a formal sum b∈B˙ cb b with all cb ∈ K. This determines an element  b∈B˙ r

∈

cb b r



SK (n, r)

r∈N

and this is clearly an element of  UK (sln ). Thus we have a map from the set of formal linear combinations of elements of B˙ to  UK (sln ). One easily checks that this is an injective algebra map. UK (sln ) arises Finally, any element (ur ) ∈   as the image of such a formal linear ˙ Write ur = combination of elements of B. b∈B˙ r cr,b b, where each cr,b ∈ K. For each b ∈ B˙ let cb = cr,b where r is the smallest value  for which π˙ n,r (b) = 0. Properties of the inverse system guarantee that the formal sum b∈B˙ cb b gives rise to (ur ) under the above map. 2 Remark 12.4. Under the identification of part (b) of the proposition, U˙ K (sln ) is the dense ˙ (Note that by 10.2, subalgebra of  UK (sln ) consisting of all finite linear combinations of B. the map πn,r restricted to UK (sln ) is still surjective.) 12.5. The map Zn → Zn−1 given by λ → λ˜ defined in 10.1 restricts to a bijection ∼ n − → Zn−1 N

n = {λ ∈ Nn | λj = 0, for some j }. Namely, if λ, µ ∈ Zn then clearly λ˜ = µ˜ if and where N only if λ and µ differ by a multiple of (1n ). So each fiber of the map Zn → Zn−1 contains n . a unique element in N

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

529

Let δ = (1n ). We have by restriction in (12.1.2) direct systems of injections AK (n, r)λ → AK (n, r + n)λ+δ → AK (n, r + 2n)λ+2δ → · · · ,

(12.5.1)

µ AK (n, r) → µ+δ AK (n, r

(12.5.2)

µ AK (n, r)λ

+ n) → µ+2δ AK (n, r + 2n) → · · · ,

→ µ+δ AK (n, r + n)λ+δ → · · ·

(12.5.3)

n such that |λ| = |µ| = r. It is clear from the preceding that their respective for any λ, µ ∈ N direct limits may be identified with K[SLn ]λ˜ , µ˜ K[SLn ], µ˜ K[SLn ]λ˜ . Similarly, we have by restriction in (12.2.1) inverse systems of surjections SK (n, r)1µ ← SK (n, r + n)1µ+δ ← SK (n, r + 2n)1µ+2δ ← · · · ,

(12.5.4)

1λ SK (n, r) ← 1λ+δ SK (n, r + n) ← 1λ+2δ SK (n, r + 2n) ← · · · ,

(12.5.5)

1λ SK (n, r)1µ ← 1λ+δ SK (n, r + n)1µ+δ ← · · ·

(12.5.6)

n such that |λ| = |µ| = r. By Lemma 2.3(b) and the isomorphism (2.4.1) for any λ, µ ∈ N these inverse systems are dual to the corresponding direct systems (12.5.1)–(12.5.3) above, so we have identifications µ˜ K[SLn ]

∗

 lim ←− SK (n, r + kn)1µ+kδ ,

(12.5.7)

K[SLn ]λ˜ ∗  lim ←− 1λ+kδ SK (n, r + kn),

(12.5.8)

µ˜ K[SLn ]λ˜

∗

 lim ←− 1λ+kδ SK (n, r + kn)1µ+kδ

(12.5.9)

for any λ, µ as above. (Here n, r are fixed.) UK (sln ) correProposition 12.6. Under the identification (12.2.2) the idempotent 1λ˜ ∈  sponds to a projection operator K[SLn ] → K[SLn ]λ˜ . Consequently, we have identifications UK (sln )1µ˜ , (a) µ˜ K[SLn ]∗   UK (sln ), (b) K[SLn ]λ˜ ∗  1λ˜  ∗ (c) µ˜ K[SLn ]λ˜  1λ˜  UK (sln )1µ˜ ˜ µ˜ ∈ Zn−1 . for any λ, Proof. From [10], the element 1λ˜ corresponds to the sequence (0, . . . , 0, 1λ , 1λ+δ , 1λ+2δ , . . .) ˜ All the in the inverse limit  UK (sln ), where λ is the element of Nn0 corresponding to λ. claims now follow from the constructions. 2

530

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

˜ µ˜ ∈ Zn−1 . Corollary 12.7. Let λ, (a) Under the identification of Proposition 12.3(b), 1λ˜  UK (sln )1λ˜ consists of all formal, ˙ ˙ ˜ = 1 ˜ U˙ K (sln )1 ˜ is the dense ˜ possibly infinite, linear combinations of B(λ), and UK (λ) λ λ ˙ λ˜ ). subalgebra consisting of all finite linear combinations of B( (b) 1λ˜  UK (sln )1λ˜  EndSLn (K[SLn ]λ˜ ); hence the generic algebra U˙ K (λ˜ ) embeds densely in an endomorphism algebra of an induced module. UK (sln )1µ˜ consists (c) Under the identification of Proposition 12.3(b), the bimodule 1λ˜  ˙ ˜ of all formal, possibly infinite, linear combinations of B(λ, µ), ˜ and the bimodule 1λ˜ U˙ K (sln )1µ˜ is the dense subspace consisting of all finite linear combinations of ˙ λ, ˜ µ). B( ˜  (d) 1λ˜ UK (sln )1λ˜  HomSLn (K[SLn ]λ˜ , K[SLn ]µ˜ ). Proof. This is immediate from the preceding proposition, the second isomorphism in (1.1.7), and the isomorphism (1.1.3) (with M = SLn ). 2

Acknowledgment The authors thank the referee for many useful suggestions.

References [1] A.A. Beilinson, G. Lusztig, R. MacPherson, A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 (1990) 655–677. [2] R.W. Carter, G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974) 193–242. [3] C. de Concini, C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976) 330–354. [4] R. Dipper, G. James, A. Mathas, Cyclotomic q-Schur algebras, Math. Z. 229 (1998) 385–416. [5] S. Donkin, On Schur algebras and related algebras I, J. Algebra 104 (1986) 310–328. [6] S. Donkin, On Schur algebras and related algebras II, J. Algebra 111 (1987) 354–364. [7] S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993) 39–60. [8] S. Doty, Resolutions of B-modules, Indag. Math. (N.S.) 5 (1994) 267–283. [9] S. Doty, Representation theory of reductive normal algebraic monoids, Trans. Amer. Math. Soc. 351 (1999) 2539–2551. [10] S. Doty, Presenting generalized q-Schur algebras, Represent. Theory 7 (2003) 196–213. [11] S. Doty, A. Giaquinto, Presenting Schur algebras, Internat. Math. Res. Notices (2002) 1907–1944. [12] S. Doty, G. Walker, Modular symmetric functions and irreducible modular representations of general linear groups, J. Pure Appl. Algebra 82 (1992) 1–26. [13] J. Du, Kazhdan–Lusztig bases and isomorphism theorems for q-Schur algebras, Contemp. Math. 139 (1992) 121–140. [14] J. Du, q-Schur algebras, asymptotic forms, and quantum SLn , J. Algebra 177 (1995) 385–408. [15] K. Erdmann, Symmetric groups and quasi-hereditary algebras, in: Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992), in: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, Kluwer, Dordrecht, 1994, pp. 123–161. [16] J.J. Graham, G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996) 1–34. [17] J.A. Green, Polynomial Representations of GLn , in: Lecture Notes in Math., vol. 830, Springer-Verlag, New York, 1980.

S. Doty et al. / Journal of Algebra 278 (2004) 502–531

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

531

J.A. Green, Combinatorics and the Schur algebra, J. Pure Appl. Algebra 88 (1993) 89–106. R.M. Green, PhD dissertation, University of Warwick, 1995. R.M. Green, Completions of cellular algebras, Comm. Algebra 27 (1999) 5349–5366. I. Grojnowski, The coproduct for quantum GLn , unpublished preprint. G.D. James, The Representation Theory of the Symmetric Groups, in: Lecture Notes in Math., vol. 682, Springer, Berlin, 1978. J.C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987. S. König, I.H. Slungård, C. Xi, Double centralizer properties, dominant dimension, and tilting modules, J. Algebra 240 (2001) 393–412. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447– 498. G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, MA, 1993. A. Mathas, Iwahori–Hecke Algebras and Schur Algebras of the Symmetric Group, in: Univ. Lecture Ser., vol. 15, Amer. Math. Soc., Providence, RI, 1999. M.S. Putcha, Linear Algebraic Monoids, in: London Math. Soc. Lecture Note Ser., vol. 133, Cambridge University Press, Cambridge, 1988. J. Rotman, Advanced Modern Algebra, Pearson Education, 2002. L. Solomon, An introduction to reductive monoids, in: Semigroups, Formal Languages and Groups (York, 1993), in: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 466, Kluwer, Dordrecht, 1995, pp. 295–352. I.D. Suprunenko, The invariance of the set of weights of irreducible representations of algebraic groups and Lie algebras of type Al with restricted weights under reduction modulo p (in Russian), Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. 2 (1983) 18–22.