Coxeter Matroids Research Articles

On T-invariant subvarieties of symplectic Grassmannians and representability of rank 2 symplectic matroids over [formula omitted

For the symplectic Grassmannian SpG(2,2n) of 2-dimensional isotropic subspaces in a 2n-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an n-dimensional torus T on it, we characterize its irreducible T-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank 2 which are representable over C.

Linear extensions and shelling orders

AbstractWe prove that linear extensions of the Bruhat order of a matroid are shelling orders and that the barycentric subdivision of a matroid is a Coxeter matroid, viewing barycentric subdivisions as subsets of a parabolic quotient of a symmetric group. A similar result holds for order ideals in minuscule quotients of symmetric groups and in their barycentric subdivisions. Moreover, we apply promotion and evacuation for labeled graphs of Malvenuto and Reutenauer to dual graphs of simplicial complexes, introducing promotion and evacuation of shelling orders.

Torus orbit closures in flag varieties and retractions on Weyl groups

A finite Coxeter group W has a natural metric d and if [Formula: see text] is a subset of W, then for each [Formula: see text], there is [Formula: see text] such that [Formula: see text]. Such q is not unique in general but if [Formula: see text] is a Coxeter matroid, then it is unique, and we define a retraction [Formula: see text] so that [Formula: see text]. The T-fixed point set [Formula: see text] of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction [Formula: see text] geometrically, where W is the Weyl group of [Formula: see text], and show that [Formula: see text]. We introduce another retraction [Formula: see text] algebraically for an arbitrary subset [Formula: see text] of W when W is a Weyl group of classical Lie type, and show that [Formula: see text] when [Formula: see text] is a Coxeter matroid.

The universal valuation of Coxeter matroids

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.

Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

Coxeter submodular functions and deformations of Coxeter permutahedra

We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This class of polytopes contains important families such as weight polytopes, signed graphic zonotopes, Coxeter matroids, root cones, and Coxeter associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.

Bruhat interval polytopes

Let u and v be permutations on n letters, with u≤v in Bruhat order. A Bruhat interval polytopeQu,v is the convex hull of all permutation vectors z=(z(1),z(2),…,z(n)) with u≤z≤v. Note that when u=e and v=w0 are the shortest and longest elements of the symmetric group, Qe,w0 is the classical permutohedron. Bruhat interval polytopes were studied recently in [15] by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety.In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and R-polynomials, we also give a generalization of the standard recurrence for R-polynomials. Finally, we define a more general class of polytopes called Bruhat interval polytopes forG/P, which are moment map images of (closures of) totally positive cells in (G/P)≥0, and are a special class of Coxeter matroid polytopes. Using tools from total positivity and the Gelfand–Serganova stratification, we show that the face of any Bruhat interval polytope for G/P is again a Bruhat interval polytope for G/P.

Brick polytopes of spherical subword complexes and generalized associahedra

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

Isotropical Linear Spaces and Valuated Delta-Matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ 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 Delta-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 $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$.

A Natural Family of Flag Matroids

A flag matroid can be viewed as a chain of matroids linked by quotients. Flag matroids, of which relatively few interesting families have previously been known, are a particular class of Coxeter matroids. In this paper we give a family of flag matroids arising from an enumeration problem that is a generalization of the tennis ball problem. These flag matroids can also be defined in terms of lattice paths, and they provide a generalization of the lattice path matroids of [J. Bonin, A. de Mier, and M. Noy, J. Combin. Theory Ser. A, 104 (2003), pp. 63–94].

Lagrangian pairs and Lagrangian orthogonal matroids

Lagrangian orthogonal matroids are the best-behaved Coxeter matroids, except for ordinary matroids, and indeed they include ordinary matroids as a special case. Represented Lagrangian orthogonal matroids arise from n -dimensional totally orthogonal subspaces of 2 n -dimensional orthogonal space, and if we take the unique pair of such subspaces which contain a given n − 1 -dimensional totally isotropic subspace, we get the prototype of a Lagrangian pair of Lagrangian orthogonal matroids. In this paper we define Lagrangian pairs, and give a number of characterizations of them and prove several properties of them. We also use them to prove a new characterization, within ordinary matroid theory, of an elementary quotient of ordinary matroids.

Symplectic matroids, independent sets, and signed graphs

We give an independent set axiomatization of symplectic matroids, a large and important class of Coxeter matroids. As an application, we construct a new class of examples of symplectic matroids from graphs. As another application, we prove that symplectic matroids satisfy a certain “basis exchange property,” and we conjecture that this basis exchange property in fact characterizes symplectic matroids.

Representations of Matroids in Semimodular Lattices

Representations of matroids in semimodular lattices and Coxeter matroids in chamber systems are considered in this paper.

Oriented Lagrangian Matroids

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural way.

Orthogonal 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 the special case where W is the symmetric group (the A n case) and P is a maximal parabolic subgroup.This generalization of matroid introduces interesting combinatorial structures corresponding to each of the finite Coxeter groups. Borovik, Gelfand and White began an investigation of the B n case, called symplectic matroids. This paper initiates the study of the D n case, called orthogonal matroids. The main result (Theorem 2) gives three characterizations of orthogonal matroid: algebraic, geometric, and combinatorial. This result relies on a combinatorial description of the Bruhat order on D n (Theorem 1). The representation of orthogonal matroids by way of totally isotropic subspaces of a classical orthogonal space (Theorem 5) justifies the terminology orthogonal matroid.

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 e 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 {\bb E}, and let The L-assignment problem is to minimize the function f_{\xi, \eta} on a given subset L ⊆ W. An important tool in proving the greedy result is a bijection between the set W/P of left cosets and a “concrete” collection A of tuples of subsets of a certain partially ordered set. If a pair of elements of W are related in the Bruhat order, then the corresponding elements of A are related in the Gale (greedy) order. Indeed, in many important cases, the Bruhat order on W is isomorphic to the Gale order on A. This bijection has an important implication for Coxeter matroids. It provides bases and independent sets for a Coxeter matroid, these notions not being inherent in the definition.

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.

Symplectic Matroids

A symplectic matroid is a collection {\cal B} of k-element subsets of J e l1, 2, …, n, 1a, 2a, … nar, each of which contains not both of i and ia for every i ≤ n, and which has the additional property that for any linear ordering p of J such that i p j implies ja p ia and i p ja implies j p ia for all i, j ≤ n, {\cal B} has a member which dominates element-wise every other member of {\cal 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 e n, and which are equivalent to Bouchet‘s symmetric matroids or 2-matroids.

On exchange properties for Coxeter matroids and oriented matroids

We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as ‘properties’ in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger than the others. The weaker ones constitute the definition of Coxeter matroids, and we also prove their equivalence to the matroid polytope property of Gelfand and Serganova.

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.