Vol. 313, No. 1, 2021

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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 = {g1,,gn} of G and every complex finite dimensional representation ρ of G, we associate the determinantal hypersurface

D(T,ρ) ={[x0 : : xn] nx 0I + xiρ(gi)  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.

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