scholarly journals K-classes for matroids and equivariant localization

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alex Fink ◽  
David Speyer
Keyword(s):  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Parallel Connection ◽  
Direct Sum ◽  
International Audience ◽  
K Theory

International audience To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult. À chaque matroïde, nous associons une classe dans la K-théorie de la grassmannienne. Nous étudions cette classe en utilisant la méthode de localisation équivariante. En particulier, nous fournissons une interprétation géométrique du polynôme de Tutte. Nous étendons également les résultats du second auteur concernant le comportement de ces classes pour la somme directe, les connexions série et parallèle et la 2-somme; ces résultats n'ont été déjà établis que pour les matroïdes réalisables, et leurs preuves précédentes étaient plus difficiles.

scholarly journals The multivariate arithmetic Tutte polynomial

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Petter Brändèn ◽  
Luca Moci
Keyword(s):  
General Framework ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Nous Proposons ◽  
International Audience

International audience We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial. Nous introduisons une version arithmétique du polynôme de Tutte multivariée récemment étudié par Sokal, et un quasi-polynôme qui interpole entre les deux. Nous proposons une représentation de Fortuin-Kasteleyn neutralise pour les matroïdes arithmétiques représentables, avec des applications aux colorations et flux arithmétiques. Nous donnons une nouvelle preuve de la positivité des coefficients du polynôme de Tutte arithmétique dans le cadre plus général des matroïdes pseudo-arithmétiques. Dans le cas d'un matroïde arithmétique représentable, nous proposons une interprétation géométrique des coefficients du polynôme de Tutte arithmétique.

Nilpotent Orbit Varieties and the Atomic Decomposition of the Q-Kostka Polynomials

1998 ◽  
Vol 50 (3) ◽  
pp. 525-537 ◽  
Author(s):  
William Brockman ◽  
Mark Haiman
Keyword(s):  
Atomic Decomposition ◽  
Jordan Block ◽  
Geometric Interpretation ◽  
Nilpotent Orbit ◽  
Cohomology Rings ◽  
Direct Sum ◽  
Kostka Polynomials ◽  
Orbit Closures ◽  
Direct Sum Decomposition ◽  
Coordinate Rings

AbstractWe study the coordinate rings of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here μ′ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous q-Kostka polynomial is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by λ in the ring . Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of as a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the , and a new proof of the atomic decomposition of the q-Kostka polynomials.

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

scholarly journals Zonotopes, toric arrangements, and generalized Tutte polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Luca Moci
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Integral Points ◽  
Poincaré Polynomial ◽  
Tutte Polynomials ◽  
International Audience ◽  
Toric Arrangement ◽  
Positive Coefficients

International audience We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope. On introduit un polynôme de Tutte avec multiplicité $M(x, y)$, qui généralise le polynôme de Tutte ordinaire et a des applications aux zonotopes et aux arrangements toriques. Nous prouvons que $M(x, y)$ satisfait une récurrence de "deletion-restriction'' et a des coefficients positifs. Le polynôme caractéristique et le polynôme de Poincaré d'un arrangement torique sont des spécialisations du polynôme associé $M(x, y)$, de même que les polynômes correspondants pour un arrangement d'hyperplans sont des spécialisations du polynôme de Tutte ordinaire. En outre, $M(1, y)$ est la série de Hilbert de l'espace discret de Dahmen-Micchelli associé, et $M(x, 1)$ calcule le volume et le nombre de points entiers du zonotope associé.

scholarly journals Monodromy and K-theory of Schubert curves via generalized jeu de taquin

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie ◽  
Jake Levinson
Keyword(s):  
Local Algorithm ◽  
Jeu De Taquin ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
The Real ◽  
Real Locus ◽  
Real Geometry ◽  
International Audience ◽  
K Theory ◽  
Rational Normal Curve

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

scholarly journals Cayley and Tutte polytopes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Matjaž Konvalinka ◽  
Igor Pak
Keyword(s):  
Complete Graph ◽  
Tutte Polynomial ◽  
Convex Hulls ◽  
Connected Graphs ◽  
Labeled Trees ◽  
Graph Traversal ◽  
International Audience ◽  
Search Graph ◽  
Traversal Algorithm

International audience Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm. Les polytopes de Cayley ont été définis récemment comme des ensembles convexes de compositions de Cayley introduits par Cayley en 1857. Dans ce papier, nous résolvons la conjecture de Braun. Cette dernière exprime le volume du polytopes de Cayley en termes du nombre de graphes connexes. Nous étendons ce résultat à des déformations de polytopes de Cayley à deux variables, à savoir les polytopes de Tutte. Le volume de ces derniers est donnè par une évaluation du polynôme de Tutte du graphe complet. Notre approche est basée sur une triangulation explicite des polytopes de Cayley et Tutte. Nous démontrons que les simplexes de ces triangulations correspondent à des arbres marqués. La pierre angulaire de notre démonstration est une bijection directe basées sur l'algorithme de la recherche du premier voisin sur le graphe.

scholarly journals On Schubert calculus in elliptic cohomology

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Kirill Zainoulline
Keyword(s):  
Formal Group ◽  
Schubert Calculus ◽  
Elliptic Cohomology ◽  
Reduced Word ◽  
Combinatorial Result ◽  
International Audience ◽  
Schubert Classes ◽  
K Theory ◽  
Nous Utilisons ◽  
Group Law

International audience An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We usethese polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective $K$-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra. Un résultat combinatoire important dans le calcul de Schubert pour la cohomologie et la $K$-théorie équivariante est représenté par les formules de Billey et Graham-Willems pour la localisation des classes de Schubert aux points fixes du tore. Ces formules sont uniformes pour tous les types de Lie, et sont basés sur le concept d’un polynôme de racines. Nous définissons les polynômes formels de racines associées à une loi arbitraire de groupe formel (et donc à une théorie de cohomologie généralisée). Nous utilisons ces polynômes pour simplifier les preuves de Billey et Graham-Willems, et aussi pour généraliser leurs résultats à la $K$-théorie connective et la cohomologie elliptique. Un autre résultat concerne la définition d’une base de Schubert dans cohomologie elliptique (c’est à dire, des classes indépendantes d’un mot réduit), en utilisant la base de Kazhdan-Lusztig de l’algèbre de Hecke correspondant.

scholarly journals Symmetries of Kirchberg Algebras

2003 ◽  
Vol 46 (4) ◽  
pp. 509-528 ◽  
Author(s):  
David J. Benson ◽  
Alex Kumjian ◽  
N. Christopher Phillips
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Direct Sum ◽  
Representation Ring ◽  
Direct Sum Decomposition ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
The Way

AbstractLet G0 and G1 be countable abelian groups. Let γi be an automorphism of Gi of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1A] = 0 in K0(A), and an automorphism α ∈ Aut(A) of order two, such that K0(A) ≅ G0, such that K1(A) ≅ G1, and such that α* : Ki(A) → Ki(A) is γi. As a consequence, we prove that every -graded countable module over the representation ring R() of is isomorphic to the equivariant K-theory K (A) for some action of on a unital Kirchberg algebra A.Along the way, we prove that every not necessarily finitely generated []-module which is free as a -module has a direct sum decomposition with only three kinds of summands, namely [] itself and on which the nontrivial element of acts either trivially or by multiplication by −1.

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