Determinants of representations of Coxeter groups

Abstract

In Ayyer et al. (J Comb Theory Ser A 150:208-232, 2017), 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.