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The q-Schur algebra. (English) Zbl 0711.20007
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
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