Determinantal hypersurfaces and representations of Coxeter groups

Let $G$ be a finitely generated group. To every generating set $T = {g_{1}, \dots, g_{n}}$ of $G$ and every complex finite dimensional representation $ρ$ of $G$ , we associate the determinantal hypersurface

D (T, ρ) = {[x_{0} : \dots : x_{n}] \in ℂ ℙ^{n} ∣ x_{0} I + \sum x_{i} ρ (g_{i}) is not invertible},

where $I$ is the identity operator, and we investigate how the geometry of this hypersurface reflects the properties of $ρ$ . 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 $D (T, ρ)$ determines $ρ$ .

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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

Authors

Željko Čučković
Department of Mathematics and Statistics University of Toledo Toledo, OH United States

Michael I. Stessin
Department of Mathematics and Statistics State University of New York, University at Albany Albany, NY United States

Alexandre B. Tchernev
Department of Mathematics and Statistics State University of New York, University at Albany Albany, NY United States

Vol. 313, No. 1, 2021

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

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors