A cancellation-free formula for the Schur elements of the Ariki-Koike algebra

TL;DR

This paper presents a new cancellation-free formula for Schur elements of the Ariki-Koike algebra, simplifying their computation and analysis in representation theory.

Contribution

It introduces a novel cancellation-free formula for Schur elements, making their factors more accessible for theoretical and computational purposes.

Findings

The formula is explicitly cancellation-free.

It facilitates easier reading and programming of Schur element factors.

Enhances understanding of the algebra's semisimplicity conditions.

Abstract

Schur elements play a powerful role in the representation theory of symmetric algebras. In the case of the Ariki-Koike algebra, Schur elements are Laurent polynomials whose factors determine when Specht modules are projective irreducible and whether the algebra is semisimple. Formulas for the Schur elements of the Ariki-Koike algebra have been independently obtained first by Geck, Iancu and Malle, and later by Mathas. The aim of this note is to give a cancellation-free formula for these polynomials, so that their factors can be easily read and programmed.