Defect in cyclotomic Hecke algebras

TL;DR

This paper proves that the defect, a key numerical invariant of blocks in cyclotomic Hecke algebras, remains invariant under block equivalences for a broad class of complex reflection groups, extending previous results.

Contribution

It establishes the defect as a block invariant for all cyclotomic Hecke algebras associated with the infinite series G(l,p,n), including type B, and introduces a new computational approach for weights.

Findings

Defect is a block invariant for all G(l,p,n) cyclotomic Hecke algebras.

For G(l,1,n), defect equals the weight defined via generalized hook lengths.

Computational evidence supports defect invariance for exceptional types.

Abstract

The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated with each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori-Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. We prove that the defect is a block invariant for all cyclotomic Hecke algebras associated with the complex reflection groups of the infinite series $G(l,p,n)$, which include the Weyl groups of type $B_n$ in the unequal parameter case. In particular, for the groups $G(l,1,n)$, we show that the defect corresponds to the notion of weight in the sense of Fayers. We thus also obtain a new way of computing the weight, which uses a generalisation of the notion of hook lengths. We further show computationally that the defect is a block invariant for all cyclotomic Hecke algebras of exceptional type for which the blocks are known, and we conjecture that the result should hold for all complex reflection groups. Finally, we obtain that the defect is also a block invariant for cyclotomic Yokonuma-Hecke algebras.