On automorphisms of undirected Bruhat graphs

The (directed) Bruhat graph $\hat{\Gamma}(u,v)$ has the elements of the Bruhat interval $[u,v]$ as vertices, with directed edges given by multiplication by a reflection. Famously, $\hat{\Gamma}(e,v)$ 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 the undirected Bruhat graph $\Gamma(e,v)$ 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 Kazhdan--Lusztig polynomials, can be characterized as certain $\Gamma(u,v)$-automorphisms which are conjecturally sufficient to generate the orbit of $e$ under $Aut(\Gamma(e,v))$.