Coxeter Complexes Research Articles

Algebraic structures on graph associahedra

M. Carr and S. Devadoss introduced in ‘Coxeter complexes and graph associahedra’ [Topology Appl. 153 (1-2) (2006) 2155–2168] the notion of tubing on a finite simple graph Γ $\Gamma$ , in the context of configuration spaces on the Hilbert plane. To any finite simple graph Γ $\Gamma$ they associated a finite partially ordered set, whose elements are the tubings of Γ $\Gamma$ and whose geometric realization is a convex polytope K Γ ${\mathcal {K}}\Gamma$ , the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for the empty graph they obtained the standard simplexes. The goal of the present work is to give an algebraic description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graph-associahedra define natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.

Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

Quantitative aspects of acyclicity

The Cheeger constant is a measure of the edge expansion of a graph and as such plays a key role in combinatorics and theoretical computer science. In recent years, there is an interest in k-dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension k. In this paper, we study several aspects of the higher Cheeger constants. Our results include methods for bounding the cosystolic norm of k-cochains and the k-th Cheeger constants, with applications to the expansion of pseudomanifolds, Coxeter complexes and homogenous geometric lattices. We revisit a theorem of Gromov on the expansion of a product of a complex with a simplex and provide an elementary derivation of the expansion in a hypercube. We prove a lower bound on the maximal cosystole in a complex and an upper bound on the expansion of bounded degree complexes and give an essentially sharp estimate for the cosystolic norm of the Paley cochains. Finally, we discuss a non-abelian version of the one-dimensional expansion of a simplex, with an application to a question of Babson on bounded quotients of the fundamental group of a random 2-complex.

Zigzag Structure of Thin Chamber Complexes

Zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all zigzags in the simplex alpha _{n}, the cross-polytope beta _{n}, the 24-cell, the icosahedron and the 600-cell are equal to the Coxeter numbers of mathsf{A}_{n}, mathsf{B}_{n}=mathsf{C}_{n}, mathsf{F}_{4} and mathsf{H}_{i}, i=3,4, respectively. We also discuss in which cases two faces in a thin chamber complex can be connected by a zigzag.

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd–Maxwell ball packings. Finally, we enumerate the Coxeter systems that generate Boyd–Maxwell ball packings.

ON -VECTORS AND THE DERIVATIVES OF THE TANGENT AND SECANT FUNCTIONS

AbstractIn this paper, we show that the$\gamma $-vectors of Coxeter complexes (of types A and B) and associahedrons (of types A and B) can be obtained by using derivative polynomials of the tangent and secant functions. We provide a unified grammatical approach to generate these$\gamma $-vectors and the coefficient arrays of Narayana polynomials, Legendre polynomials and Chebyshev polynomials of both kinds.

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler–Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, ( q , p ) -analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.

Connectivity of chamber graphs of buildings and related complexes

Let Δ be a thick and locally finite building with the property that no edge of the associated Coxeter diagram has label “ ∞ ”. The chamber graph G ( Δ ) , whose edges are the pairs of adjacent chambers in Δ , is known to be q -regular for a certain number q = q ( Δ ) . Our main result is that G ( Δ ) is q -connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of Δ is connected by q pairwise disjoint galleries. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.

On γ-Vectors Satisfying the Kruskal–Katona Inequalities

We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative.

On $\gamma$-vectors satisfying the Kruskal-Katona inequalities

We present examples of flag homology spheres whose $\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose $f$-vectors are the $\gamma$-vectors in question, and so a result of Frohmader shows that the $\gamma$-vectors satisfy not only the Kruskal-Katona inequalities but also the stronger Frankl-Füredi-Kalai inequalities. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+3$ vertices its $\gamma$-vector satisfies the Frankl-Füredi-Kalai inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma (\Delta)$ satisfies the Kruskal-Katona, and further, the Frankl-Füredi-Kalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$-vectors are nonnegative. Nous présentons des exemples de sphères d'homologie flag dont $\gamma$-vecteurs satisfaire les inégalités de Kruskal-Katona. Cela comprend plusieurs familles de bien étudiés simplicial complexes, y compris les complexes de Coxeter et les complexes simpliciaux dual de l'associahedron et à la cyclohedron. Dans ces cas, nous construisons explicite flag simplicial complexes dont les $f$-vecteurs sont les $\gamma$-vecteurs en question, et ainsi de suite de Frohmader montre que le $\gamma$-vecteurs de satisfaire non seulement les inégalités de Kruskal-Katona mais aussi les plus fortes inégalités Frankl-Füredi-Kalai. Dans une autre direction, nous montrons que, si un flag $(d-1)$-sphère a au plus $2d+3$ sommets son $\gamma$-vecteur satisfait aux inégalités de Frankl-Füredi-Kalai. Nous conjecture que, si $\Delta$ est une sphère d'homologie flag alors $\gamma(\Delta)$ satisfait aux inégalités de Kruskal-Katona, en outre, celles de Frankl-Füredi-Kalai. Cette conjecture est un raffinement significatif de la conjecture de Gal, qui affirme que ces $\gamma$-vecteurs sont nonnégatifs.

The Dynkin diagram cohomology of finite Coxeter groups

Let D be a connected graph. The Dynkin complex C D ( A ) of a D-algebra A was introduced by the second author to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering the Dynkin complex by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W.

Shelling Coxeter-like complexes and sorting on trees

In their work on ‘Coxeter-like complexes’, Babson and Reiner introduced a simplicial complex Δ T associated to each tree T on n nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that Δ T is ( n − b − 1 ) -connected when the tree has b leaves. We provide a shelling for the ( n − b ) -skeleton of Δ T , thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree T which imply shellability of Δ T , and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes M m , n with n ⩾ 2 m − 1 . We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.

On certain combinatorial expansions of the Eulerian polynomials

We study in this work properties of a combinatorial expansion of the classical Eulerian polynomial A n ( t ) , including the recurrence relation and the exponential generating function for the expansion coefficients. Analogous results for the Eulerian polynomials of types B and D are also obtained. The expansions obtained enable us to readily deduce the symmetry, unimodality, and alternating behavior at t = − 1 of the corresponding Eulerian polynomials, where the latter property settles the Charney–Davis conjecture for the Coxeter complexes of types A, B and D with combinatorial interpretations given to the corresponding Charney–Davis quantities.

Coxeter complexes and graph-associahedra

Given a graph Γ, we construct a simple, convex polytope, dubbed graph-associahedra, whose face poset is based on the connected subgraphs of Γ. This provides a natural generalization of the Stasheff associahedron and the Bott–Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne–Knudsen–Mumford compactification of the real moduli space of curves.

Coxeter-like complexes

Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.

On Modular Homology of Simplicial Complexes: Saturation

Among shellable complexes a certain class is shown to have maximal modular homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated.

Metric characterizations of spherical and Euclidean buildings

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.

Flag-transitive Extensions of Dual Affine Planes

In this paper I finish the work carried on in [10] and [1], obtaining a classification of flag-transitive geometries belonging to the following diagram: (with 1<q<∞).An infinite family of simply connected examples for this diagram with q=2is obtained by truncating Coxeter complexes of type D n. Two different finite families with three members and q=4arise from the Steiner systems for the Mathieu groups M 2 2, M 2 3 and M 2 4. The geometries of these two families are simply connected. Three more simply connected examples of rank 3 are described in [1], with q=2, 4and 16 respectively. We shall prove that no more simply connected examples exist besides those mentioned above.

On Flat Flag-Transitive c.c^* -Geometries

We study flat flag-transitive c.c^{\ast}-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of then -dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism groupG = 2^{2n}\cdot L_n(2) and covered by the truncated Coxeter complex of typeD_{2^n} . The non-canonical ways give us geometries with smaller automorphism group (G \le 2^{2n}\cdot (2^n-1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.

H-Shellings and h-Complexes

We introduce and study a new class of shellings of simplicial complexes that we call h-shellings. These shellings allow the construction of a subcomplex of the original one with particularly nice enumerative and topological properties. Several infinite classes of such complexes are discussed. Included in these classes are Coxeter complexes, Tits buildings, and order complexes of many shellable posets.