The q-Schur algebra. (English) Zbl 0711.20007

Dipper, Richard; James, Gordon

Proc. Lond. Math. Soc., III. Ser. 59, No. 1, 23-50 (1989).

The authors introduce and study the q-Schur algebra, which plays for the Hecke algebra of type A an analogous role to that played by the classical Schur algebra for the symmetric group. Let R be an integral domain and q a unit in R. Let \(\sigma_ k\) denote the symmetric group on k symbols and \(H_ R\) the Hecke algebra associated with \(\sigma_ k\) over R. Then \(H_ R\) has a standard basis \(\{T_ w|\) \(w\in \sigma_ k\}\). Let \(\lambda\) be a partition of k (written \(\lambda\vdash k)\) and \(W_{\lambda}\) the standard Young subgroup of \(\sigma_ k\) corresponding to \(\lambda\). Let \(x_{\lambda}=\sum_{w\in W_{\lambda}}T_ w\). Then the q-Schur algebra \(S_ R(q,k)\) is defined as \(S_ R(q,k)=End_{H_ R}(\oplus_{\lambda \vdash k}x_{\lambda}H_ R)\). In order to study the modular representation theory of the finite general linear group \(GL_ n(q)\) the authors then take \(R=K\), \({\mathcal O}\) or F where (F,\({\mathcal O},K)\) is a split p-modular system for \(GL_ n(q)\), p being a prime which does not divide q. Suppose \(n=dk\) and s is a non-zero root of a monic irreducible polynomial of degree d over \(F_ q\). Then s corresponds to a conjugacy class of semisimple elements with exactly one elementary divisor over \(F_ q\), in \(GL_ n(q)\). By earlier work of the second author [Proc. Lond. Math. Soc., III. Ser. 52, 236-268 (1986; Zbl 0587.20022)] there is a lower unitriangular matrix \(\Delta\) (s,k) associated with s, whose rows and columns are indexed by partitions of k, and which is a part of the p- modular decomposition matrix of \(GL_ n(q)\). Furthermore, the authors showed [J. Algebra 104, 266-288 (1986; Zbl 0622.20032)] how the complete p-modular decomposition matrix can be calculated from a knowledge of the \(\Delta\) (s,k) for various s of degree d over \(F_ q\) and dk\(\leq n\). The main results of this paper (Theorem 4.9) relates the decomposition numbers of the \(q^ d\)-Schur algebra with the matrix \(\Delta\) (s,k) with \(dk=n\). More precisely, with a suitable ordering of the partitions of k, the decomposition numbers \(\tilde d_{\lambda \mu}\) of the \(q^ d\)- Schur algebra (\(\lambda\),\(\mu\vdash k)\) are related to the decomposition numbers \(d_{\lambda \mu}\) appearing in \(\Delta\) (s,k) by \(\tilde d_{\lambda \mu}=d_{\lambda '\mu '}\) (here \(\lambda '\) is the partition dual to \(\lambda\)). It follows then that for primes p not dividing q, the p-modular decomposition matrices of q-Schur algebras determine the p-modular decomposition matrix of \(GL_ n(q)\). In §6 the authors show that the p-modular decomposition matrices of q-Schur algebras also determine the decomposition numbers of \(GL_ n(F)\) where F is a sufficiently large field of characteristic p. If \(k\leq n\) and \(\lambda\vdash k\) let \(W^{\lambda}\), \(L^{\lambda}\) denote a Weyl module and an irreducible module of \(GL_ n(F)\). They show (Theorem 6.9) that if p divides \(q^ d-1\), the decomposition number \(\tilde d_{\lambda \mu}\) of the \(q^ d\)-Schur algebra equals the composition multiplicity of \(L^{\mu}\) in \(W^{\lambda}\). In §7 they give a complete recipe for calculating the decomposition numbers of \(GL_ n(q)\) from a knowledge of the decomposition matrices of Schur algebras. They also give a new proof of a theorem of P. Fong and the reviewer [Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)] which classifies the p-blocks of \(GL_ n(q)\). Some numerical examples are given in §5.

Reviewer: B.Srinivasan

MSC:

20C20	Modular representations and characters
20C30	Representations of finite symmetric groups
20G05	Representation theory for linear algebraic groups
20G40	Linear algebraic groups over finite fields

Keywords:

Hecke algebra; symmetric group; modular representation; finite general linear group; split p-modular system; semisimple elements; complete p- modular decomposition matrix; decomposition numbers; partitions; q-Schur algebras; Weyl module; irreducible module; composition multiplicity; p- blocks