Spectral analysis and representations]{Spectral analysis near regular point of reducibility and representations of Coxeter groups

TL;DR

This paper develops a local spectral analysis near regular points of reducibility for determinantal hypersurfaces and applies it to prove a rigidity theorem for Coxeter group representations.

Contribution

It introduces a new local spectral analysis framework for determinantal hypersurfaces and establishes a rigidity theorem for Coxeter group representations.

Findings

Spectral analysis near regular reducibility points is developed.

A rigidity theorem for Coxeter group representations is proved.

The approach links spectral properties with algebraic group representations.

Abstract

For a tuple of square matrices $A_1,...,A_n$ the determinantal hypersurface is defined as \begin{eqnarray*} &\sigma(A_1,...,A_n)= \\ &\Big\{[x_1:\cdots :x_n]\in \C{\mathbb P}^{n-1}: det(x_1A_1+\cdots +x_nA_n)=0\Big \}. \end{eqnarray*} In this paper we develop a local spectral analysis near a regular point of reducibility of a determinantal hypersurface. We prove a rigidity type theorem for representations of Coxeter groups as an application