Hook formula for Coxeter groups via the twisted group ring

TL;DR

This paper generalizes Nakada's colored hook formula for Coxeter groups using Kostant and Kumar's twisted group ring, providing new proofs and extending the formula's applicability to Kac--Moody algebra Weyl groups.

Contribution

It introduces a novel approach using twisted group rings to generalize and prove Nakada's colored hook formula for all Coxeter groups.

Findings

Generalization of Nakada's colored hook formula to all Coxeter groups

A new proof for the formula in the case of dominant minuscule Weyl group elements

Extension of the formula's applicability to Kac--Moody algebra Weyl groups

Abstract

We use Kostant and Kumar's twisted group ring and its dual to formulate and prove a generalization of Nakada's colored hook formula for any Coxeter groups. For dominant minuscule elements of the Weyl group of a Kac--Moody algebra, this provides another short proof of Nakada's colored hook formula.