Morita equivalences of cyclotomic Hecke algebras of type $G(r,p,n)$ II: the $(\epsilon ,q)$-separated case

Jun Hu and Andrew Mathas

Abstract

The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with $(\epsilon ,q)$-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type $G(s,1,m)$, where $1\le s\le r$ and $1\le m\le n$. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers meaning that the decomposition matrices of these algebras are now known in principle. In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type $G(r,p,n)$ when the parameters are $(\epsilon ,q)$-separated. The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the $l$-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type $G(r,p,n)$.

Keywords: Hecke algebra, decomposition matrices, complex reflection groups.

AMS Subject Classification: Primary 20C08; secondary 20C30.