Determinantal hypersurfaces and representations of Coxeter groups

Željko Čučković, Michael I. Stessin and Alexandre B. Tchernev

Vol. 313 (2021), No. 1, 103–135

DOI: 10.2140/pjm.2021.313.103

Abstract

Let $G$ be a finitely generated group. To every generating set $T=\left\{g_1,\dots ,g_n\right\}$ of $G$ and every complex finite dimensional representation $\rho$ of $G$, we associate the determinantal hypersurface

$$D\left(T,\rho \right)=\left\{\left[x_0:\cdots :x_n\right]\in ℂ{\mathbb{P}}^n\mid x_{0}I+\sum x_i\rho \left(g_i\right)\text{ is not invertible }\right\},$$

where $I$ is the identity operator, and we investigate how the geometry of this hypersurface reflects the properties of $\rho$. In the classical case when $G$ is a finite Coxeter group of regular type and $T$ is a Coxeter generating set for $G$ we show that $D\left(T,\rho \right)$ determines $\rho$.

Keywords

determinantal hypersurfaces, Coxeter groups, joint spectrum, group representations.

Mathematical Subject Classification

Primary: 20F55, 20C33

Secondary: 47A67, 14J70

Milestones

Received: 11 July 2020

Revised: 7 May 2021

Accepted: 8 June 2021

Published: 17 September 2021