A $\lowercase{q}$-deformation of a trivial symmetric group action

Let $\mathcal{A}$ be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system $A_{n-1}$. Let $B=B(q)$ be the Varchenko matrix for this arrangement with all hyperplane parameters equal to $q$. We show that $B$ is the matrix with rows and columns indexed by permutations with $\sigma, \tau$ entry equal to $q^{i(\sigma \tau^{-1})}$ where $i(\pi)$ is the number of inversions of $\pi$. Equivalently $B$ is the matrix for left multiplication on $\mathbb{C}\mathfrak{S}_n$ by

Clearly $B$ commutes with the right-regular action of $\mathfrak{S}_n$ on $\mathbb{C}\mathfrak{S}_n$. A general theorem of Varchenko applied in this special case shows that $B$ is singular exactly when $q$ is a $j(j-1)^{st}$ root of $1$ for some $j$ between $2$ and $n$. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the $\mathfrak{S}_n$-module structure of the nullspace of $B$ in the case that $B$ is singular. Our first result is that

in the case that $q = e^{2\pi i/n(n-1)}$ where Lie$_n$ denotes the multilinear part of the free Lie algebra with $n$ generators. Our second result gives an elegant formula for the determinant of $B$ restricted to the virtual $\mathfrak{S}_n$-module $P^\eta$ with characteristic the power sum symmetric function $p_\eta(x)$.