An Adjacency Criterion for Coxeter Matroids

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A _n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.     

Boundaries of Coxeter Matroids

The purpose of this paper is to introduce, for a finite Coxeter groupW, the mod 2 boundary operator on the space of all Coxeter matroids (also known asWP-matroids) forWandP, wherePvaries through all the proper standard parabolic subgroups ofW(Theorem 3 of the paper). A remarkably simple interpretation of Coxeter matroids as certain sets of faces of the generalized permutahedron associated with the Coxeter groupW(Theorem 1) yields a natural definition of the boundary of a Coxeter matroid. The latter happens to be a union of Coxeter matroids for maximal standard parabolic subgroupsQ_iofP(Theorem 2). These results have very natural interpretations in the case of ordinary matroids and flag-matroids (Section 3).     

The Greedy Algorithm and Coxeter Matroids

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = A (isomorphic to the symmetric group Sym_{_n+1}) and P a maximal parabolic subgroup. The main result of this paper is that for Coxeter matroids, just as for ordinary matroids, the greedy algorithm provides a solution to a naturally associated combinatorial optimization problem. Indeed, in many important cases, Coxeter matroids are characterized by this property. This result generalizes the classical Rado-Edmonds and Gale theorems.A corollary of our theorem is that, for Coxeter matroids L, the greedy algorithm solves the L-assignment problem. Let W be a finite group acting as linear transformations on a Euclidean space , and let

A geometric characterization of Coxeter matroids

Coxeter matroids, introduced by Gelfand and Serganova, are combinatorial structures associated with any finite Coxeter group and its parabolic subgroup they include ordinary matroids as a specia case. A basic result in the subject is a geometric characterization of Coxeter matroids first stated by Gelfand and Serganova. This paper presents a self-contained, simple proof of a more general version of this geometric characterization.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

Symplectic Matroids

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i n, and which has the additional property that for any linear ordering of J such that i j implies j* i* and i j* implies j i* for all i, j n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids.     

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.     

Isotropical linear spaces and valuated Delta-matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n×n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Δ-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

Strong Reality Properties of Normalizers of Parabolic Subgroups in Finite Coxeter Groups

Let W be a finite Coxeter group, P a parabolic subgroup of W, and N _W (P) the normalizer of P in W. We prove that every element in N _W (P) is strongly real in N _W (P), and that every irreducible complex character of N _W (P) has Frobenius-Schur indicator 1.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

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.     

Reflection Independence in Even Coxeter Groups

If (W,S) is a Coxeter system, then an element of W is a reflection if it is conjugate to some element of S. To each Coxeter system there is an associated Coxeter diagram. A Coxeter system is called reflection preserving if every automorphism of W preserves reflections in this Coxeter system. As a direct application of our main theorem, we classify all reflection preserving even Coxeter systems. More generally, if (W,S) is an even Coxeter system, we give a combinatorial condition on the diagram for (W,S) that determines whether or not two even systems for W have the same set of reflections. If (W,S) is even and (W,S) is not even, then these systems do not have the same set of reflections. A Coxeter group is said to be reflection independent if any two Coxeter systems (W,S) and (W,S) have the same set of reflections. We classify all reflection independent even Coxeter groups.Mathematics Subject Classifications (2000). 20F05, 20F55, 20F65, 51F15.     

Atomic domains in which almost all atoms are prime

ABSTRACT

Let W be a (finite or infinite) Coxeter group and W _X be a proper standard parabolic subgroup of W. We show that the semilattice made up by W equipped with the weak order is a semidirect product of two smaller semilattices associated with W _X .

     

On the Topology of the Combinatorial Flag Varieties

We prove that the simplicial complex Ω _n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of W -matroids for an arbitrary finite Coxeter group W , and generalises the well-known results by Solomon—Tits and Bj?rner on spherical buildings. Received January 16, 2000, and in revised form October 7, 2000, and April 16, 2001. Online publication December 21, 2001.     

A subalgebra of 0-Hecke algebra

Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

Coxeter–Chein Loops

In 1974, Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group W together with ?₂ by a variation on a semidirect product. We first settle an open problem, originally proposed by Petr Vojtěchovský in 2003, by finding a minimal presentation for the Chein loop with respect to a presentation for W. We then study these loops in the case where W is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. In particular, even if the Moufang loop is infinite, it is finitely presented. Viewing these presentations as amalgams of loops, we then apply methods due to Blok and Hoffman to describe a family of twisted Coxeter–Chein loops.