Determinants of representations of Coxeter groups

Abstract

In [APS17], 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 ω of W , we give a closed formula for the number of irreducible representations of W with determinant ω. For Coxeter groups of type B n and D n , this is accomplished by characterizing the bipartitions associated to such representations.

Macdonald trees and determinants of representations for finite Coxeter groups

“…This article is mainly a survey of the main results of three papers [1] and [2] and [7] concerning the application of the dyadic arithmetic of partitions to the representation theory of finite Coxeter groups.…”

“…This is used to explain the results for all other Coxeter groups, which are taken from Ref. [7], and appear in later subsections. 3,5,4,8,12,20,8,16,24,40,32,64,88,152,16,32,48, .…”

“…The determinants of the representations of B n can be computed either through the Solomon formulas, combined with the Frobenius character formula, or through a well-known formula for the determinant of an induced representation. (See [7], for example.) Let…”

“…For n as in (1), let n = n − . The following equations compute N sgn 0 (n) in all cases [7]. Let k = v 2 (n ).…”

“…The dihedral case is facile, and the exceptional cases are finite in number and the determinants may be computed by available character tables. See [7] for these remaining cases. Moreover, a(n)/b(n + 2) = 1 if and only if n is divisible by 4.…”