Elsevier

Advances in Mathematics

Volume 374, 18 November 2020, 107389
Advances in Mathematics

Separable elements and splittings of Weyl groups

https://doi.org/10.1016/j.aim.2020.107389Get rights and content

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×UW 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.

Introduction

A permutation w=w1wn is separable if it avoids the patterns 3142 and 2413, meaning that there are no indices i1<i2<i3<i4 such that the values wi1wi2wi3wi4 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 W/U in a Coxeter group W, where UW is an arbitrary subset:W/U:={wW|(wu)=(w)+(u),uU}. They proved that W/U is always an interval [e,w0u01]L in the left weak order, where u0 is the least upper bound of U in the right weak order. When U=WJ is a parabolic subgroup, the generalized quotient W/U is precisely the parabolic quotient WJ. It is well known that the multiplication map WJ×WJW is a length-additive bijection. Any such pair (X,Y) of subsets of W for which the multiplication map X×YW 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 W/[e,u]R×[e,u]RW when u is separable, answering an open problem of Wei [17]. Next, in the case W=Sn, we show in Theorem 4 that any splitting X×YW 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 W/[e,u]R×[e,u]RW is surjective for any uW=Sn. 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 uW and nested sets Nu 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 [e,u]L and [e,u]R in terms of the nested set Nu; 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 Φ+. We freely use the well-known Cartan-Killing classification of irreducible root systems into types An,Bn,Cn,Dn,G2,F4,E6,E7, and E8, 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 wW(Φ). Then w is separable if one of the following holds:

  • (S1)

    Φ is of type A1;

  • (S2)

    Φ=Φi is reducible and w|Φi 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:W/U={wW|(wu)=(w)+(u),uU}.

Proposition 6

Björner and Wachs [4]

Let u0=uURu, then W/U=[e,w0u01]L.

A pair (X,Y) of arbitrary subsets X,YW such that the multiplication map X×YW sending (x,y)xy is length-additive (meaning (xy)=(x)+(y),xX,yY) and bijective is called a splitting of W. Generalized quotients generalize the notion of parabolic quotients, since WJ=W/WJ; Proposition 3 implies that we have a splitting (WJ,WJ) 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 2r copies of the graph associahedron A(Γ) 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 αα whenever sα and sα 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 (X,Y) be a splitting of a Weyl group W, then:

  • (1)

    X and Y have unique maximal elements x0 and y0 under left and right weak order respectively. Furthermore, we have x0y0=w0.

  • (2)

    Let JΔ, then (XWJ,YWJ) is a splitting of WJ.

Proof

By the definition of splitting, there exist unique elements x0X and y0Y such that x0y0=w0. Since all products xy with xX and y

Proof of Theorem 5

The proof of Theorem 5 relies on the following technical lemma.

Lemma 5

Let w,π,uSn such that uLw and uRπ. If(wu1)+(u)+(u1π)>(wu1π) then there exists u>Bu in the strong Bruhat order such that uLw and uRπ.

We now observe that Theorem 5 follows from Lemma 5.

Proof of Theorem 5

For any x,ySn and any maximal element z in the Bruhat order such that zLx and zRy, we have(xz1)+(z)+(z1y)=(xz1y), since otherwise a strictly larger z can be found via Lemma 5. In particular, there is at least one element zSn

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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C.G. is partially supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1122374.

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