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The Bergman complex of a matroid and phylogenetic trees
 the Journal of Combinatorial Theory, Series B. arXiv:math.CO/0311370
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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 equivalen ..."
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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. 2 The terminology in the present paper follows [BG, BR] (though we prefer to use the name `Coxeter matroids' rather than `WP matroids,' as used in these papers); see also the forthcoming book [BGW1]. The cited publications also contain all the necessary background material. For more detail, refer to books [We],[Wh], [O] and [R] for the systematic exposition of matroid theory and theory of Coxeter complexes. The authors wish to thank A. Kelmans for several helpful suggestions. 1 Exchange properties for matroids Matroids. The following is wellknown (see for example [O]): Theorem 1.1 Let B be a nonempty collection of subsets of E. Then the following are equivalent: (1) For every A, B # B and a # A \ B there exists b # B \ A such that A \ {a} # {b} # B (the Exchange Property). (2) For every A, B # B and a # A \ B there exists b # B \ A such that B \ {b} # {a} # B (the Dual Exchange Property). (3) For every A, B # B and a # A \ B, there exists b # B \ A such that A\{a}#{b} # B and B \{b}#{a} # B (the Symmetric Exchange Property).
A quasisymmetric function for matroids
"... Abstract. A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating ..."
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Abstract. A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, • is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight, • behaves simply under matroid duality, • has a simple expansion in terms of Ppartition enumerators, and • is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle
POSITROID VARIETIES I: JUGGLING AND GEOMETRY
, 903
"... ABSTRACT. While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposi ..."
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ABSTRACT. While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, and BrownGoodearlYakimov. However, its cyclicinvariance is hidden in this description. Postnikov gave many cyclicinvariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We adopt his terminology and call the strata positroid varieties. We show that positroid varieties are normal and CohenMacaulay, and are defined as schemes by the vanishing of Plücker coordinates. We compute their Tequivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and BuchKreschTamvakis ’ approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the HodgeGröbner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective StanleyReisner scheme of a shellable ball. CONTENTS
Matroid Polytopes and Their Volumes
"... Abstract. We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the ..."
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Abstract. We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra. Résumé. On exprime le polytope matroïde PM d’un matroïde M comme somme signée de Minkowski de simplices, et on obtient une formule pour le volume de PM. Ceci donne une expression combinatoire pour le degré d’une clôture d’orbite de tore dans la Grassmannienne Grk,n. Ensuite, on deduit des résultats analogues pour le polytope ensemble indépendant et pour le polytope matroïde drapeau associé a M. Nos preuves sont fondées sur une extension naturelle de la théorie de Postnikov de permutoèdres généralisés.
Lectures on matroids and oriented matroids
"... Let’s begin with a little “pep talk”, some (very) brief history, and some of the motivating examples of matroids. 1.1. Motivation. Why learn about or study matroids/oriented matroids in geometric, topological, algebraic combinatorics? Here are a few of my personal reasons. • They are general, so res ..."
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Let’s begin with a little “pep talk”, some (very) brief history, and some of the motivating examples of matroids. 1.1. Motivation. Why learn about or study matroids/oriented matroids in geometric, topological, algebraic combinatorics? Here are a few of my personal reasons. • They are general, so results about them are widely applicable. • They have relatively few axioms and standard constructions/techniques, so they focus one’s approach to solving a problem. • They give examples of wellbehaved objects: polytopes, cell/simplicial complexes, rings. • They provide “duals ” for nonplanar graphs! 1.2. Brief early history. (in no way comprehensive...) 1.2.1. Matroids. • H. Whitney (1932, 1935) graphs, duality, and matroids as abstract linear independence.
The Bergman Complex of a Matroid and Phylogenetic Trees
"... Abstract. We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. ..."
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Abstract. We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan e B(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn. 1.