Abstract:For a Coxeter system $(W,S)$ 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 $M_n$.
From: Alexandru Chirvăsitu L. [view email]
[v1]
Mon, 26 May 2025 01:17:52 UTC (37 KB)
[v2]
Sat, 22 Nov 2025 17:39:17 UTC (39 KB)
[v3]
Fri, 26 Dec 2025 12:36:05 UTC (24 KB)
[v4]
Fri, 28 Aug 2026 15:49:38 UTC (24 KB)
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