Decomposition numbers for Hecke algebras of type $G(r,p,n)$: the $(\epsilon,q)$-separated case

TL;DR

This paper develops a comprehensive framework for calculating decomposition numbers of cyclotomic Hecke algebras of type G(r,p,n) with separated parameters, linking them to simpler cases and providing explicit algorithms.

Contribution

It introduces a Specht module theory, constructs simple modules, and establishes Morita equivalences to compute decomposition matrices explicitly for these algebras.

Findings

Decomposition numbers are determined by related G(s,1,m) algebras.

An explicit algorithm for computing decomposition numbers is provided.

Decomposition matrices are known in characteristic zero.

Abstract

The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with $(\eps,q)$-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. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero. 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 $(\eps,q)$-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 \textit{$l$-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 $G(r,p,n)$.