Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over ?[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of “quasiparabolic” subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.     

Shortest path poset of Bruhat intervals

We define the shortest path poset SP(u,v) of a Bruhat interval [u,v], by considering the shortest u–v paths in the Bruhat graph of a Coxeter group W, where u,v∈W. We consider the case of SP(u,v) having a unique rising chain under a reflection order and show that in this case SP(u,v) is a Gorenstein^? poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.     

Maps between buildings that preserve a given Weyl distance

Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω ε W, we study surjective maps ? : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (?(C), ?(D)) = ω. The result is that the restrictions of ? to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ? is an isomorphism. In particular, ? is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S.     

Monotonicity of strata in the stratification of the cone of totally positive matrices

According to a theorem of Bjorner [5], there exists a stratified space whose strata are labeled by the elements of [u, v] for every interval [u, v] in the Bruhat order of a Coxeter group W, and each closed stratum (respectively the boundary of each stratum) has the homology of a ball (respectively of a sphere). In [6], Fomin and Shapiro suggest a natural geometric realization of these stratified spaces for a Weyl group W of a semi-simple Lie group G, and then prove its validity in the case of the symmetric group. The stratified spaces arise as links in the Bruhat decomposition of the totally non-negative part of the unipotent radical of G. In this article, we verify the topological regularity property of the strata formed as a result of Bruhat partial ordering on the elements of theWeyl group (of rank 4) of a semi-simple simply connected algebraic group G which is SL(4,?) in our case here. The Weyl group here is the Coxeter group S ₄.     

P-kernels,IC bases and Kazhdan–Lusztig polynomials

In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W.     

On the Fully Commutative Elements of Coxeter Groups

Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide. Partially supported by NSF Grants DMS-9057192 and DMS-9401575.     

Inversion arrangements and Bruhat intervals

Let W be a finite Coxeter group. For a given w∈W, the following assertion may or may not be satisfied:

(?)

The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w.

We present a type independent combinatorial criterion which characterises the elements w∈W that satisfy (?). A couple of immediate consequences are derived:

(1)

The criterion only involves the order ideal of w as an abstract poset. In this sense, (?) is a poset-theoretic property.

(2)

For W of type A, another characterisation of (?), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result.

(3)

If W is a Weyl group and the Schubert variety indexed by w∈W is rationally smooth, then w satisfies (?).

     

Partial vexillarity and bigrassmannian permutations

We define two closely related notions of degree for permutation patterns of type 2143. These give rise to classes of “m-vexillary elements” in the symmetric group. Using partitions, the Ehresmann–Bruhat partial order, and sets constructed from permutation inversions, we characterize the m-vexillary elements. We relate the maximal bigrassmannian permutations in the (Ehresmann–Bruhat) order ideal generated by any given m-vexillary element w to the maximal rectangles contained in the shape of w.     

Antichain Cutsets of Strongly Connected Posets

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected d-uniform hypergraphs.     

Fixed points of involutive automorphisms of the Bruhat order

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein^* over Z₂ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right.     

Parking spaces

Let W be a Weyl group with root lattice Q and Coxeter number h  . The elements of the finite torus Q/(h+1)Q$Q/(h+1)Q$ are called the W-parking functions, and we call the permutation representation of W on the set of W-parking functions the (standard) W-parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W-parking spaces, called the noncrossing parking space and the algebraic parking space, with the following features:

•

They are defined more generally for real reflection groups.     

Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schrödinger operator with quasi-periodic potential v_n=λv(x+nω). We assume that the frequency ω satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove—in the perturbative regime—that for large disorder λ and for most frequencies ω the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.     

Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

Parking structures: Fuss analogs

For any irreducible real reflection group W with Coxeter number h, Armstrong, Reiner, and the author introduced a pair of \(W \times {\mathbb {Z}}_{h}\) -modules which deserve to be called W-parking spaces which generalize the type A notion of parking functions and conjectured a relationship between them. In this paper we give a Fuss analog of their constructions. For a Fuss parameter k≥1, we define a pair of \(W \times {\mathbb {Z}}_{kh}\) -modules which deserve to be called k-W-parking spaces and conjecture a relationship between them. We prove the weakest version of our conjectures for each of the infinite families ABCDI of finite reflection groups, together with proofs of stronger versions in special cases. Whenever our weakest conjecture holds for W, we have the following corollaries.

• There is a simple formula for the character of either k-W-parking space.
• We recover a cyclic sieving result due to Krattenthaler and Müller which gives the cycle structure of a generalized rotation action on k-W-noncrossing partitions.
• When W is crystallographic, the restriction of either k-W-parking space to W is isomorphic to the action of W on the finite torus Q/(kh+1)Q, where Q is the root lattice.

     

Cambrian lattices

For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a “cluster fan.” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.     

The <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">ω</Emphasis></Superscript>-normalizers of finite groups

Given a nonempty set ω of primes and a nonempty formation F of finite groups, we define the F ^ω -normalizer in a finite group and study their properties (existence, invariance under certain homomorphisms, conjugacy, embedding, and so on) in the case that F is an ω-local formation. We so develop the results of Carter, Hawkes, and Shemetkov on the F-normalizers in groups.     

Noncrossing partitions and the shard intersection order

We define a new lattice structure \((W,\preceq)\) on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC?(W) as a sublattice. The new construction of NC?(W) yields a new proof that NC?(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of \((W,\preceq)\), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC?(W). There is a natural dimension-preserving bijection between simplices in the order complex of \((W,\preceq)\) (i.e. chains in \((W,\preceq)\)) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC?(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron.The lattice \((W,\preceq)\) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.     

On intersections of conjugacy classes and bruhat cells

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B ^– ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB ^– is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution m _C ∈ 2 W associated to C. We prove that the element m _C is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe m _C explicitly for every conjugacy class C, and when w ∈ W ≌ S_n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.