The many graded cellular bases of Hecke algebras

TL;DR

This paper constructs numerous graded cellular bases for Hecke algebras of complex reflection groups, enabling explicit module construction and proving key conjectures in the representation theory of these algebras.

Contribution

It introduces new graded cellular bases for Hecke algebras, resolving a longstanding problem and enabling explicit module construction and conjecture proofs.

Findings

Constructed many graded integral cellular bases

Explicitly constructed simple modules of Ariki's categorification

Proved Martin-Woodcock's conjecture

Abstract

We settle a long-standing problem in the theory of Hecke algebras of complex reflection groups by constructing many (graded) integral cellular bases of these algebras. As applications, we explicitly construct the simple modules of Ariki's categorification theorem and prove unitriangularity of decomposition matrices over arbitrary fields, we also prove Martin-Woodcock's conjecture.