Separable elements and splittings of Weyl groups

Abstract

We continue the study of separable elements in finite Weyl groups, introduced in [11]. These elements generalize the well-studied class of separable permutations. We show that the multiplication map W/U×U→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, where W/U denotes the generalized quotient. This generalizes a result for the symmetric group, answering an open problem of Wei [17].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 [4]. 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 [13]. 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.