On automorphisms of undirected Bruhat graphs

Abstract

The undirected Bruhat graph \(\Gamma (u,v)\) has the elements of the Bruhat interval [u, v] as vertices, with edges given by multiplication by a reflection. Famously, \(\Gamma (e,v)\) is regular if and only if the Schubert variety \(X_v\) is smooth, and this condition on v is characterized by pattern avoidance. In this work, we classify when \(\Gamma (e,v)\) is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of \(\Gamma (u,v)\) in the course of which we show that special matchings, which originally appeared in the theory of Kazhdan–Lusztig polynomials, can be characterized, for the symmetric and right-angled groups, as certain \(\Gamma (u,v)\)-automorphisms which are conjecturally sufficient to generate the orbit of e under \({{\,\textrm{Aut}\,}}(\Gamma (e,v))\).