Morita equivalence for blocks of Hecke algebras of symmetric groups (Q1365018)

This paper deals with the Hecke algebras which arise as endomorphism rings of the permutation representation of the finite general linear group on the cosets of the Borel subgroup which can also be viewed as specialization of the generic Hecke algebra \({\mathcal H}_q(S_n)\) of the symmetric group \(S_n\). Connections between representations of \({\mathcal H}_q(S_n)\) and \(S_n\) have been developed by \textit{R. Dipper} and \textit{G. James} [Proc. Lond. Math. Soc., III. Ser. 52, 20-52 (1986; Zbl 0587.20007) and ibid. 54, 57-82 (1987; Zbl 0615.20009)] and this paper reveals more of those connections. In particular it was shown that the bases of \(q\)-Specht modules and \(q\)-permutation modules for \({\mathcal H}_q(S_n)\) are \(q\)-analogues of the branching of the bases of the corresponding Specht and permutation modules for \(S_n\). Also a \(q\)-analogue of the branching theorem of Specht modules is proved for \(q\)-Specht modules over an arbitrary field. The main theme of this paper is to generalize a result proved by \textit{J. Scopes} [J. Algebra 142, No. 2, 441-455 (1991; Zbl 0736.20008)] on Morita equivalence of blocks of symmetric groups. The notion of \((w,k)\)-pair of blocks of \(S_n\) is generalized for blocks of \({\mathcal H}_{R,q}(S_n)\) and it is proved that blocks of \({\mathcal H}_{F,q}(S_n)\) forming a \((w,k)\)-pair have the same decomposition matrix.