Decomposition matrices for $d$-Harish-Chandra series: the exceptional rank two cases

TL;DR

This paper computes the decomposition matrices for cyclotomic Hecke algebras associated with rank 2 exceptional complex reflection groups in characteristic 0, establishing foundational representation-theoretic results.

Contribution

It provides explicit decomposition matrices and proves the existence of canonical basic sets for these algebras, advancing understanding of their modular representations.

Findings

All decomposition matrices for the specified algebras are calculated.

Canonical basic sets are proven to exist in the sense of Geck-Rouquier.

Modular irreducible representations can be lifted to ordinary representations.

Abstract

We calculate all decomposition matrices of the cyclotomic Hecke algebras of the rank 2 exceptional complex reflection groups in characteristic 0. We prove the existence of canonical basic sets in the sense of Geck-Rouquier and show that all modular irreducible representations can be lifted to the ordinary ones.