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.