  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.
