Representations of Hecke algebras of type \(B_n\)
In previous papers [see e.g. Proc. Lond. Math. Soc., III. Ser. 52, 20-52 (1986; Zbl 0587.20007)] the authors studied the representations of Hecke algebras of type \(A\). In this paper they study the Hecke algebra \({\mathcal H}(W_ n)\), where \(W_ n\) is the Weyl group of type \(B_ n\), over a commutative ring \(R\). The algebra \({\mathcal H}(W_ n)\) is generated by elements \(T_ t\), \(T_{s_ i}\) \((i = 1, 2, \dots,n)\) with relations involving two elements \(q\), \(Q \in R\), where \(q\) is assumed to be a unit in \(R\) and the subalgebra generated by the \(T_{s_ i}\) is isomorphic to \({\mathcal H}(S_ n)\) where \(S_ n\) is the symmetric group. If \(a \leq n\) is a non-negative integer let \(S_{(a, n - a)}\) denote the Young subgroup of \(S_ n\) fixing all \(i\) with \(1 \leq i \leq a\). Assume now that \(R\) is an integral domain. Let \(f_ n (Q,q) = \prod^{n - 1}_{i = -(n - 1)}\) \((Q + q^ i)\). The first main result of this paper is that if \(f_ n (Q + q)\) is a unit in \(R\), the category of \({\mathcal H}(W_ n)\)- modules is Morita equivalent to the category of modules for the ring \(\bigoplus^ n _{a = 0} {\mathcal H}(S_{(a, n - a)})\). The authors then give an explicit way of obtaining the simple \({\mathcal H}(W_ n)\)-modules from those of the \({\mathcal H}(S_{(a, n - a)})\) by using certain idempotents \(e_{a, n - a}\). In [loc. cit.] they constructed Specht modules \(S^ \lambda\) of \({\mathcal H}(S_ n)\), where \(\lambda \vdash n\). Assuming further that \(R\) is a field, they describe a complete set of non-isomorphic simple \({\mathcal H}(W_ n)\)-modules \(D^{\lambda, \mu}\) in terms of Specht modules \(S^ \lambda\), \(S^ \mu\) of \({\mathcal H}(S_ a)\), \({\mathcal H}(S_{n - a})\) respectively. Here \(\lambda \vdash a\), \(\mu \vdash n - a\), and \(\lambda\), \(\mu\) are \(e\)-regular where \(e\) is the least positive integer \(r\) such that \(1 + q + \cdots + q^{r - 1} \neq 0\). They then classify the blocks of \({\mathcal H}(W_ n)\), and show that the decomposition matrix can be obtained from the decomposition matrices of the various \({\mathcal H}(S_ a)\).
scientific article; zbMATH DE number 34222
scientific article