#### Vol. 313, No. 1, 2021

 Recent Issues Vol. 317: 1 Vol. 316: 1  2 Vol. 315: 1  2 Vol. 314: 1  2 Vol. 313: 1  2 Vol. 312: 1  2 Vol. 311: 1  2 Vol. 310: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
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 $T\phantom{\rule{-0.17em}{0ex}}=\left\{{g}_{1},\dots ,{g}_{n}\right\}$ of $G$ and every complex finite dimensional representation $\rho$ of $G$, we associate the determinantal hypersurface

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