Generalised hook lengths and Schur elements for Hecke algebras

TL;DR

This paper explores generalized hook lengths in partitions and their impact on Schur elements within Hecke algebras, revealing divisibility and product relations that deepen understanding of modular representation theory.

Contribution

It introduces new generalizations of hook lengths and demonstrates their application to Schur elements in Ariki-Koike and type A Hecke algebras, establishing divisibility and product formulas.

Findings

Schur element of a simple module is divisible by that of its core.

In type A, Schur element equals the product of core and quotient Schur elements.

Provides new insights into the structure of Hecke algebra representations.

Abstract

We compare two generalisations of the notion of hook lengths for partitions. We apply this in the context of the modular representation theory of Ariki-Koike algebras. We show that the Schur element of a simple module is divisible by the Schur element of the associated (generalised) core. In the case of Hecke algebras of type $A$, we obtain an even stronger result: the Schur element of a simple module is equal to the product of the Schur element of its core and the Schur element of its quotient.