Separable elements and splittings of Weyl groups

Abstract

We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or splitting, of the Weyl group $W$ when $U$ is an order ideal in right weak order generated by a separable element; this generalizes a result for the symmetric group, answering an open problem of Wei. For a generalized quotient of the symmetric group, we show that this multiplication map is a bijection if and only if $U$ is an order ideal in right weak order generated by a separable element, thereby classifying those generalized quotients which induce splittings of the symmetric group, resolving a problem of Björner and Wachs from 1988. We also prove that this map is always surjective when $U$ is an order ideal in right weak order. Interpreting these sets of permutations as linear extensions of 2-dimensional posets gives the first direct combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem Morales, Pak, and Panova. We also prove a new $q$-analog of Sidorenko's formula. All of these results are conjectured to extend to arbitrary finite Weyl groups. Finally, we show that separable elements in $W$ are in bijection with the faces of all dimensions of several copies of the graph associahedron of the Dynkin diagram of $W$. This correspondence associates to each separable element $w$ a certain nested set; we give product formulas for the rank generating functions of the principal upper and lower order ideals generated by $w$ in terms of these nested sets, generalizing several known formulas.