Spectral selections, commutativity preservation and Coxeter-Lipschitz maps
Alexandru Chirvasitu
TL;DR
The paper investigates maps that preserve spectral and commutativity structure, connecting operator-theoretic rigidity with Coxeter-group combinatorics. It develops a circle-configuration and Weyl-group framework to classify continuous, spectrum- and commutativity-preserving maps on ${\mathrm{SU}}(n)$, showing each restricts on maximal tori to either conjugation (with potential transpose) or a diagonal spectral selection, yielding a unitary analogue of Petek–HM. In parallel, it proves a strong rigidity result for right-$T$-Lipschitz self-maps on finitary, connected Coxeter systems: such maps are either constant or right translations, with a precise description in terms of projections onto connected components. Together, these results illuminate how Coxeter geometry governs spectral-preserving maps and extend known Hermitian-case classifications to the full matrix setting. The work thus connects spectral- and commutativity-preserving matrix maps with Coxeter/Lipschitz dynamics, offering a unifying perspective and concrete classification results.
Abstract
Let $(W,S)$ be a Coxeter system whose graph is connected, with no infinite edges. A self-map $τ$ of $W$ such that $τ_{σθ}\in \{τ_θ,\ στ_θ\}$ for all $θ\in W$ and all reflections $σ$ (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 $σ$ range over smaller, $θ$-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 $M_n$.