Separable elements and splittings of Weyl groups
Introduction
A permutation is separable if it avoids the patterns 3142 and 2413, meaning that there are no indices such that the values are in the same relative order as 3142 or 2413. This well-studied class of permutations arose in the study of pop-stack sorting [1] and has found applications in algorithmic pattern matching and bootstrap percolation [7], [15]. These permutations have a remarkable recursive combinatorial structure and are enumerated by the Schröder numbers [18].
This paper2 is a sequel to [11]. Whereas that paper was concerned with defining separable elements in arbitrary finite Weyl groups and establishing some of their structural properties (such as their characterization by root system pattern avoidance) this paper is concerned with certain algebraic decompositions of the Weyl group induced by separable elements and with applying results about these decompositions to resolve several open problems. In addition, we show that the combinatorics of separable elements is closely linked with the combinatorics of graph associahedra. Most of our results are new even for the case of the symmetric group.
Björner and Wachs [4] introduced the notion of a generalized quotient in a Coxeter group W, where is an arbitrary subset: They proved that is always an interval in the left weak order, where is the least upper bound of U in the right weak order. When is a parabolic subgroup, the generalized quotient is precisely the parabolic quotient . It is well known that the multiplication map is a length-additive bijection. Any such pair of subsets of W for which the multiplication map is a length-additive bijection is called a splitting.
In Section 2 we recall background on Weyl groups, root systems, and the weak order which is not specific to the study of separable elements.
Section 3 defines the notion of a separable element in a finite Weyl group W and states some results from [11] which will be needed later.
Section 4 states our three main results about generalized quotients. First, in Theorem 3 we show that there is a splitting when u is separable, answering an open problem of Wei [17]. Next, in the case , we show in Theorem 4 that any splitting is of this form; this solves a problem of Björner and Wachs [4] from 1988. Lastly, in Theorem 5 we show that the multiplication map is surjective for any . Together with the discussion in Section 4.1, this resolves an open problem of Morales, Pak, and Panova [13]. In Section 4.1 we also give a new q-analog of an inequality for linear extensions of 2-dimensional posets due to Sidorenko [16]. In Section 4.2 we conjecture that Theorem 4, Theorem 5 extend to arbitrary finite Weyl groups.
In Section 5 we give an elegant bijection between separable elements and nested sets on Γ, the Dynkin diagram associated to W. By a result of Postnikov [14], these nested sets index the faces of the graph associahedron of Γ. We give a product formula for the rank generating functions of and in terms of the nested set ; this formula generalizes several formulas in the literature.
Finally, Sections 6 and 7 contain the proofs of Theorem 4, Theorem 5 respectively.
Section snippets
Background and definitions
This section consists of background and definitions relating to root systems, Weyl groups, and the weak and strong Bruhat orders; all of this material is standard and may be found, for example, in [3].
Throughout the paper, Φ will denote a finite, crystallographic root system with chosen set of simple roots Δ and corresponding set of positive roots . We freely use the well-known Cartan-Killing classification of irreducible root systems into types , and , although all
Separable elements of Weyl groups
We now introduce a definition of a separable element in any finite Weyl group. This definition coincides exactly with separable permutations in the case of the symmetric group, although this is only made clear by Theorem 2 below, where separable elements are characterized by root system pattern avoidance. Theorem 7 in Section 5 gives another characterization of separable elements.
Definition 1 Let . Then w is separable if one of the following holds: Φ is of type ; is reducible and is
Generalized quotients and splittings of Weyl groups
Given any subset U of a Weyl group W, Björner and Wachs [4] introduced the generalized quotient:
Proposition 6 Let , then .Björner and Wachs [4]
A pair of arbitrary subsets such that the multiplication map sending is length-additive (meaning ) and bijective is called a splitting of W. Generalized quotients generalize the notion of parabolic quotients, since ; Proposition 3 implies that we have a splitting in
Product formulas and graph associahedra
In this section we show that separable elements in W are in bijection with the faces of all dimensions of copies of the graph associahedron of the Dynkin diagram Γ for W, where W has r irreducible factors. The Dynkin diagram is a graph with vertices indexed by the simple roots Δ and edges whenever and do not commute; we often identify subgraphs of Γ with the corresponding subsets of Δ when convenient. It is well-known that all connected components of Dynkin diagrams of
Proof of Theorem 4
In Proposition 9, Proposition 10 we give several methods of producing more splittings from a given one; these will be useful in the proof of Theorem 4.
Proposition 9 Let be a splitting of a Weyl group W, then: X and Y have unique maximal elements and under left and right weak order respectively. Furthermore, we have . Let , then is a splitting of .
Proof
By the definition of splitting, there exist unique elements and such that . Since all products xy with and
Proof of Theorem 5
The proof of Theorem 5 relies on the following technical lemma. Lemma 5 Let such that and . If then there exists in the strong Bruhat order such that and .
We now observe that Theorem 5 follows from Lemma 5.
Proof of Theorem 5 For any and any maximal element z in the Bruhat order such that and , we have since otherwise a strictly larger z can be found via Lemma 5. In particular, there is at least one element
Acknowledgments
We wish to thank Anders Björner for alerting us to important references and Vic Reiner, Richard Stanley, and Igor Pak for helpful comments. We are especially grateful to our advisor Alex Postnikov for his observation that separable elements may be related to faces of graph associahedra.
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