2025/05/26 by Alexandru Chirvasitu, Chirvasitu, Alexandru
Mathematics · #06A06 #15B57 #20B30 #20F10 #20F55 #47A10 #54C05 #54H15 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Mathematical Analysis and Transform Methods #Metric Geometry (math.MG) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2505.19393
openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (W,S) be a Coxeter system whose graph is connected, with no infinite edges. A self-map τ of W such that τσθ∈ \τθ, στθ\ for all θ∈ 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 Sn, 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 SU(n)→ Mn 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 Hn≤ Mn of self-adjoint matrices, strengthening it slightly by expanding the codomain to Mn.