Determinantal hypersurfaces and representations of Coxeter groups

Given a finite generating set $T=\{g_0,\dots, g_n\}$ of group $G$, and representation $\rho$ $G$ on Hilbert space $V$, we investigate how the geometry $D(T,\rho)=\{ [x_0 : \dots x_n] \in\mathbb C\mathbb P^n \mid \sum x_i\rho(g_i) \text{ not invertible} \}$ reflects properties $\rho$. When $V$ is finite-dimensional this an algebraic hypersurface in $\mathbb P^n$. In special case $T=G$ $\rho=$ left regular defined by \emph{group determinant}, object studied extensively founding work Frobenius that lead to creation theory. We focus classic when Coxeter group, make $T$ adding identity element $1_G$ for $G$. Under these assumptions show our first main result if representation, then $D(T,\rho)$ determines isomorphism class Our second exceptional type, any dimensional