Cyclotomic \(q\)-Schur algebras
Recently, the authors of the present article [Proc. Lond. Math. Soc., III. Ser. 77, No. 2, 327-361 (1998; Zbl 0916.20006)] and, independently \textit{J. Du} and \textit{L. Scott} [Trans. Am. Math. Soc. (to appear)], defined an analogue of the \(q\)-Schur algebra for an Iwahori-Hecke algebra of type \(B\). In this paper the authors study an analogue of the \(q\)-Schur algebra for an arbitrary Ariki-Koike algebra. An Ariki-Koike algebra \(H\) is a cyclotomic algebra of type \(G(r,1,n)\) and becomes an Iwahori-Hecke algebra of type \(A\) or \(B\) when \(r=1\) or \(2\), respectively. For each multicomposition \(\lambda\) of \(n\), a right ideal \(M^\lambda\) of \(H\) is constructed. The cyclotomic \(q\)-Schur algebra is then defined to be \(S=\text{End}_H(\oplus_\lambda M^\lambda)\). A Specht series for \(M^\lambda\) is then obtained and this is used to construct a cellular basis of \(S\). As a consequence, a complete set of non-isomorphic irreducible \(S\)-modules is obtained, and it is also shown that the cyclotomic \(q\)-Schur algebra is quasi-hereditary.
scientific article; zbMATH DE number 1336831