Determinants of Representations of Coxeter Groups

TL;DR

This paper extends the characterization of representations with nontrivial determinants from symmetric groups to all finite Coxeter groups, providing formulas and classifications for types B and D.

Contribution

It introduces a general formula for counting irreducible representations of Coxeter groups with a given determinant, expanding previous results beyond symmetric groups.

Findings

Closed formula for the number of irreducible representations with a given determinant.

Characterization of bipartitions for types B and D Coxeter groups.

Extension of previous symmetric group results to all finite Coxeter groups.

Abstract

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial multiplicative character $\omega$ of $W$, we give a closed formula for the number of irreducible representations of $W$ with determinant $\omega$. For Coxeter groups of type $B_n$ and $D_n$, this is accomplished by characterizing the bipartitions associated to such representations.