Let (W, S) be a Coxeter system whose graph is connected, with no infinite edges. A self-map of W such that _ \_{, \ _\} for all W and all reflections (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 range over smaller, -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.
Alexandru Chirvasitu (Sun,) studied this question.