Spectral selections, commutativity preservation and Coxeter-Lipschitz maps

TL;DR

This paper characterizes certain spectrum- and commutativity-preserving maps on special unitary groups and self-adjoint matrices, revealing they are essentially conjugations, transpositions, or spectrum selections, based on combinatorial properties of Coxeter systems.

Contribution

It establishes a link between combinatorial properties of Coxeter systems and the structure of spectrum-preserving maps on matrix groups, extending previous results.

Findings

Maps are conjugations, transpositions, or spectrum selections.

Results apply to continuous spectrum- and commutativity-preserving maps.

Extends Petek's results from self-adjoint matrices to all matrices.

Abstract

Let $(W,S)$ be a Coxeter system whose graph is connected, with no infinite edges. A self-map $\tau$ of $W$ such that $\tau_{\sigma\theta}\in \{\tau_{\theta},\ \sigma\tau_{\theta}\}$ for all $\theta\in W$ and all reflections $\sigma$ (analogous to being 1-Lipschitz with respect to the Bruhat order on $W$) is either constant or a right translation. A somewhat stronger version holds for $S_n$, where it suffices that $\sigma$ range over smaller, $\theta$-dependent sets of reflections. These combinatorial results have a number of consequences concerning continuous spectrum- and commutativity-preserving maps $\mathrm{SU}(n)\to M_n$ defined on special unitary groups: every such map is a conjugation composed with (a) the identity; (b) transposition, or (c) a continuous diagonal spectrum selection. This parallels and recovers Petek's analogous statement for self-maps of the space $H_n\le M_n$ of self-adjoint matrices, strengthening it slightly by expanding the codomain to $M_n$.