Decomposition matrices for the generic Hecke algebras on 3 strands in characteristic 0

TL;DR

This paper classifies all decomposition matrices of certain generic Hecke algebras associated with exceptional complex reflection groups on 3 strands in characteristic zero, showing all ordinary representations are modular reductions of irreducible ones.

Contribution

It provides a complete classification of decomposition matrices for these Hecke algebras, a new result in the representation theory of complex reflection groups.

Findings

All ordinary representations are obtained as modular reductions.

Complete classification of decomposition matrices for G_4, G_8, G_{16}.

Results hold for all parameter choices in characteristic zero.

Abstract

We classify all the decomposition matrices of the generic Hecke algebras on 3 strands in characteristic 0. These are the generic Hecke algebras associated to the exceptional complex reflection groups $G_4$, $G_8$ and $G_{16}$. We prove that for every choice of the parameters that define these algebras, all ordinary representations are obtained as modular reductions of irreducible representations.