DOCSLIB.ORG

Tropical Linear Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tropical Linear Algebra Tropical Linear Algebra Jorge Alberto Olarte February 18, 2019 Joint work with Alex Fink, Benjamin Schr¨oter and Marta Panizzut Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 1 / 14 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. 3 By its Plucker¨ coordinates. Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 3 By its Plucker¨ coordinates. Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 Classical linear spaces Let K be any field. A d-dimensional linear subspace L ⊆ Kn can be given in several forms: d 1 As a span of vectors d vectors v1;:::; vd 2 K . This can be represented by a d×n matrix A 2 K where the rows are given by the vectors v1;:::; vd . 2 As the solution set of n − d linear equations. This can be represented by a matrix A? 2 K(n−d)×n where rows give the coefficients of the linear equations. 3 By its Plucker¨ coordinates. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 2 / 14 (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 Classical Plucker¨ coordinates [n] d×d For any B 2 d (a subset of size d of f1;:::; ng) let AB 2 K be the submatrix of A consisting of the columns indexed by B. Definition (n)−1 The Plucker¨ coordinates of L consist in a vector in PK d whose entries are the maximal minors [det(AB )] [n] of any matrix A whose rows form a basis of L. B2( d ) This does not depend on the choice of basis. (n)−1 A vector W 2 PK d is the vector of Plucker¨ coordinates of a linear space if and only if it satisfies the Plucker¨ relations: [n] [n] X 8S 2 ; 8T 2 ; W · W = 0 d − 1 d + 1 T ni S[i i2T nS The linear space L can be recovered from the Plucker¨ coordinates: ( ) [n] X L = x 2 n j 8T 2 ; W · x = 0 K d + 1 T ni i i2T Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 3 / 14 The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 The tropical semiring The tropical semiring is T = (R [ f1g; ⊕; ; 1; 0) where a ⊕ b = min(a; b) a b = a + b The tropical projective space is TPn−1 = (Tn n f(1;:::; 1)g) = ∼ where (x1;:::; xn) ∼ (y1;:::; yn) if and only if there exists λ 2 T such that (x1;:::; xn) = (y1 λ, : : : ; yn λ) A tropical polynomial is of the form M α f (x1;:::; xn) = cα x α2∆ = min (cα + α1 · x1 + ··· + αn · xn) α2∆ n where the coefficients cα 2 T are tropical numbers and ∆ is a finite subset of N . The tropical analogous of the 'zeros' of f consists of al x 2 Tn where the minimum in f is either 1 or achieved twice. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 4 / 14 Tropical linear spaces Definition (n)−1 A valuated matroid W is a vector in PT d that satisfies the 3-term tropical [n] Plucker¨ relations: for every S 2 d−2 and i; j; k; l 2 [n] n S, the minimum of (WSij WSkl ) ⊕ (WSij WSkl ) ⊕ (WSij WSkl ) is achieved twice. The tropical linear space associated to W is ( ) [n] M L := x 2 n j 8T 2 ; the minimum W · x is achieved twice T d + 1 T ni i i2T Tropical linear spaces are polyhedral complexes. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 5 / 14 Matroid polytopes Definition n The hypersimplex ∆d;n is the intersection of unit hypercube [0; 1] with the hyperplane fx1 + ··· + xn = dg.A matroid polytope M is a subpolytope of ∆d;n such that all of its edges are also edges of ∆d;n The hypersimplex ∆2;4 This square pyramid is This symplex is NOT a is an octahedron. matroid polytope. matroid polytope. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 6 / 14 Regular subdivisions Definition Given a polytope P with vertex set V and a height function h : V ! R the lower faces of conv(f(v; h(v)) 2 Rn+1 j v 2 V g) project onto P to form a polyhedral subdivision Sub(h). Such subdivisions are called regular. This gives RV a fan structure Sec(P) called the secondary fan of P, which consists of cones σ(S) = fh 2 RV j Sub(h) = Sg for each regular subdivision S. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 7 / 14 The tropical linear space L associated to W is dual to the subcomplex of Sub(W) which consists of all polytopes of Sub(W) that are not contained in any of the coordinate hyperplanes fxi = 0g Matroid subdivisions Theorem (Speyer) (n)−1 A vector W 2 PT d is a valuated matroid if and only if the polytope conv(feB j WB < 1g) is a matroid polytope and the regular subdivision Sub(W) consists of only matroid polytopes. Jorge Alberto Olarte Tropical Linear Algebra February 18, 2019 8 / 14 Matroid subdivisions Theorem (Speyer) (n)−1 A vector W 2 PT d is a valuated matroid if and only if the polytope conv(feB j WB < 1g) is a matroid polytope and the regular subdivision Sub(W) consists of only matroid polytopes.
Load more

Recommended publications
  • Geometry of Generalized Permutohedra
    Geometry of Generalized Permutohedra by Jeffrey Samuel Doker A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Federico Ardila, Co-chair Lior Pachter, Co-chair Matthias Beck Bernd Sturmfels Lauren Williams Satish Rao Fall 2011 Geometry of Generalized Permutohedra Copyright 2011 by Jeffrey Samuel Doker 1 Abstract Geometry of Generalized Permutohedra by Jeffrey Samuel Doker Doctor of Philosophy in Mathematics University of California, Berkeley Federico Ardila and Lior Pachter, Co-chairs We study generalized permutohedra and some of the geometric properties they exhibit. We decompose matroid polytopes (and several related polytopes) into signed Minkowski sums of simplices and compute their volumes. We define the associahedron and multiplihe- dron in terms of trees and show them to be generalized permutohedra. We also generalize the multiplihedron to a broader class of generalized permutohedra, and describe their face lattices, vertices, and volumes. A family of interesting polynomials that we call composition polynomials arises from the study of multiplihedra, and we analyze several of their surprising properties. Finally, we look at generalized permutohedra of different root systems and study the Minkowski sums of faces of the crosspolytope. i To Joe and Sue ii Contents List of Figures iii 1 Introduction 1 2 Matroid polytopes and their volumes 3 2.1 Introduction . .3 2.2 Matroid polytopes are generalized permutohedra . .4 2.3 The volume of a matroid polytope . .8 2.4 Independent set polytopes . 11 2.5 Truncation flag matroids . 14 3 Geometry and generalizations of multiplihedra 18 3.1 Introduction .
    [Show full text]
  • Matroid Polytopes and Their Volumes Federico Ardila, Carolina Benedetti, Jeffrey Doker
    Matroid Polytopes and Their Volumes Federico Ardila, Carolina Benedetti, Jeffrey Doker To cite this version: Federico Ardila, Carolina Benedetti, Jeffrey Doker. Matroid Polytopes and Their Volumes. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.77-88. hal-01185426 HAL Id: hal-01185426 https://hal.inria.fr/hal-01185426 Submitted on 20 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2009, Hagenberg, Austria DMTCS proc. AK, 2009, 77–88 Matroid Polytopes and Their Volumes Federico Ardila1,y Carolina Benedetti2 zand Jeffrey Doker3 1San Francisco, CA, USA, [email protected]. 2Universidad de Los Andes, Bogota,´ Colombia, [email protected]. 3University of California, Berkeley, Berkeley, CA, USA, [email protected]. 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.
    [Show full text]
  • Isotropical Linear Spaces and Valuated Delta-Matroids
    Journal of Combinatorial Theory, Series A 119 (2012) 14–32 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta Isotropical linear spaces and valuated Delta-matroids Felipe Rincón University of California, Berkeley, Department of Mathematics, 1045 Evans Hall, Berkeley, CA, United States article info abstract Article history: The spinor variety is cut out by the quadratic Wick relations Received 3 June 2010 among the principal Pfaffians of an n × n skew-symmetric matrix. Availableonline27August2011 Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this Keywords: picture, and we develop a combinatorial theory of tropical Wick Tropical linear space Isotropic subspace vectors and tropical linear spaces that are tropically isotropic. We Delta matroid characterize tropical Wick vectors in terms of subdivisions of - Coxeter matroid matroid polytopes, and we examine to what extent the Wick Valuated matroid relations form a tropical basis. Our theory generalizes several Spinor variety results for tropical linear spaces and valuated matroids to the class Wick relations of Coxeter matroids of type D. Matroid polytope © 2011 Elsevier Inc. All rights reserved. Tropical basis 1. Introduction Let n be a positive integer, and let V be a 2n-dimensional vector space over an algebraically closed field K of characteristic 0. Fix a basis e1, e2,...,en, e1∗ , e2∗ ,...,en∗ for V , and consider the symmetric bilinear form on V defined as n n Q (x, y) = xi yi∗ + xi∗ yi, i=1 i=1 for any two x, y ∈ V with coordinates x = (x1,...,xn, x1∗ ,...,xn∗ ) and y = (y1,...,yn, y1∗ ,...,yn∗ ).
    [Show full text]
  • Arxiv:0905.4405V1 [Math.CO] 27 May 2009
    Matroid Polytopes: Algorithms, Theory, and Applications By DAVID CARLISLE HAWS B.S. (University of California, Davis) 2004 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Jes´usA. De Loera (chair) Matthias K¨oppe arXiv:0905.4405v1 [math.CO] 27 May 2009 Eric Babson Committee in Charge 2009 -i- c David Carlisle Haws, 2009. All rights reserved. Contents Abstract iii Acknowledgments iv Chapter 1. What are Matroids and who are their polytopes?1 1.1. Matroids1 1.2. Matroid Polytopes & Polymatroids7 Chapter 2. Volume and Ehrhart Polynomial Computation 17 2.1. Computing the Ehrhart Polynomials 18 2.2. Preliminaries on Rational Generating Functions. 18 2.3. On the Tangent Cones of Matroid Polytopes 20 2.4. Polymatroids 30 2.5. The construction of a short multivariate rational generating function 32 2.6. Polynomial-time specialization of rational generating functions in varying dimension 35 Chapter 3. Results on the Algebraic Combinatorics of Matroid Polytopes 43 3.1. Algebraic Properties of h∗-vectors and Ehrhart polynomials of Matroid Polytopes 43 3.2. Unimodular Simplices of Matroid Polytopes 53 3.3. Two Dimensional Faces of Matroid Polytopes 64 Chapter 4. Applications to Optimization Through the Structure of Matroid Polytopes 67 4.1. Description of the Algorithms and Heuristics 67 4.2. Software Implementation 79 4.3. Computational Results 80 Bibliography 126 -ii- Matroid Polytopes: Algorithms, Theory and Applications. Abstract This dissertation presents new results on three different themes all related to matroid polytopes.
    [Show full text]
  • Ehrhart Polynomials of Matroid Polytopes and Polymatroids
    EHRHART POLYNOMIALS OF MATROID POLYTOPES AND POLYMATROIDS JESUS´ A. DE LOERA, DAVID C. HAWS, AND MATTHIAS KOPPE¨ Abstract. We investigate properties of Ehrhart polynomials for matroid poly- topes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are com- putable in polynomial time. The proof relies on the geometry of these poly- topes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h∗-vector and the co- efficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity. 1. Introduction Recall that a matroid M is a finite collection F of subsets of [n] = {1, 2, . , n} called independent sets, such that the following properties are satisfied: (1) ∅ ∈ F, (2) if X ∈ F and Y ⊆ X then Y ∈ F, (3) if U, V ∈ F and |U| = |V | + 1 there exists x ∈ U \V such that V ∪x ∈ F. In this paper we investigate convex polyhedra associated with matroids. One of the reasons matroids have become fundamental objects in pure and ap- plied combinatorics is their many equivalent axiomatizations. For instance, for a matroid M on n elements with independent sets F the rank function is a func- tion ϕ: 2[n] → Z where ϕ(A) := max{ |X| | X ⊆ A, X ∈ F }. Conversely a function ϕ: 2[n] → Z is the rank function of a matroid on [n] if and only if the following are satisfied: (1) 0 ≤ ϕ(X) ≤ |X|, (2) X ⊆ Y =⇒ ϕ(X) ≤ ϕ(Y ), (3) ϕ(X ∪ Y ) + ϕ(X ∩ Y ) ≤ ϕ(X) + ϕ(Y ).
    [Show full text]
  • Matroid Polytopes and Their Volumes
    matroid polytopes and their volumes. federico ardila∗ carolina benedettiy jeffrey dokerz 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 inde- pendent 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. 1 introduction. The theory of matroids can be approached from many different points of view; a matroid can be defined as a simplicial complex of independent sets, a lattice of flats, a closure relation, etc. A relatively new point of view is the study of matroid polytopes, which in some sense are the natural combinatorial incarnations of matroids in algebraic geometry and optimization. Our paper is a contribution in this direction. We begin with the observation that matroid polytopes are members of the family of generalized permutohedra [14]. With some modifications of Postnikov's beautiful theory, we express the matroid polytope PM as a signed Minkowski sum of simplices, and use that to give a formula for its volume Vol (PM ). This is done in Theorems 2.5 and 3.3. Our answers are expressed in terms of the beta invariants of the contractions of M. Formulas for Vol (PM ) were given in very special cases by Stanley [17] and Lam and Postnikov [11], and a polynomial time algorithm for finding Vol (PM ) ∗San Francisco State University, San Francisco, CA, USA, [email protected].
    [Show full text]
  • Isotropical Linear Spaces and Valuated Delta-Matroids Felipe Rincón
    Isotropical Linear Spaces and Valuated Delta-Matroids Felipe Rincón To cite this version: Felipe Rincón. Isotropical Linear Spaces and Valuated Delta-Matroids. 23rd International Confer- ence on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.801-812. hal-01215074 HAL Id: hal-01215074 https://hal.inria.fr/hal-01215074 Submitted on 13 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2011, Reykjav´ık, Iceland DMTCS proc. AO, 2011, 801–812 Isotropical Linear Spaces and Valuated Delta-Matroids Felipe Rincon´ 1y 1University of California, Berkeley, CA, USA Abstract. 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 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.
    [Show full text]
  • Cyclic Polytopes and Oriented Matroids
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2000) 21, 49–64 Article No. eujc.1999.0317 Available online at http://www.idealibrary.com on Cyclic Polytopes and Oriented Matroids RAUL CORDOVIL† AND PIERRE DUCHET Consider the moment curve in the real euclidean space Rd defined parametrically by the map d 2 d γ R R , t γ .t/ .t; t ;:::; t /. The cyclic d-polytope d .t1;:::; tn/ is the convex hull of Vn >!d different7! pointsD on this curve. The matroidal analogs areC the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that of .t ;:::; tn/: We give combinatorial and geometrical characterizations of cyclic (matroid) poly- Cd 1 topes. A simple evenness criterion determining the facets of d .t1;:::; tn/ was given by Gale [21]. We characterize the admissible orderings of the vertices of theC cyclic polytope, i.e., those linear or- derings of the vertices for which Gale’s evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes. c 2000 Academic Press 1. INTRODUCTION The standard dth cyclic polytope with n vertices (where n > d), denoted by d .t1; t2;:::, C tn/, was discovered by Caratheodory´ [8, 9] in the context of harmonic analysis, and has been rediscovered many times; it is defined as the convex hull in Rd of n > d different points d 2 d γ .t1/; : : : ; γ .tn/ on the moment curve γ R R , t γ .t/ .t; t ;:::; t /.
    [Show full text]
  • Valuative Invariants for Polymatroids
    FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 271–282 Valuative invariants for polymatroids Harm Derksen1y and Alex Fink2 1University of Michigan 2University of California, Berkeley Abstract. Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia- Reiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z-modules of all Z-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the Z-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that G is universal for valuative invariants. Resum´ e.´ Beaucoup des invariants importants des matro¨ıdes et polymatro¨ıdes, tels que le polynomeˆ de Tutte, la fonc- tion quasi-symmetrique de Billera-Jia-Reiner, et l’invariant G introduit par le premier auteur, sont valuatifs. Dans cet article nous construisons les Z-modules de fonctions valuatives aux valeurs entieres` des matro¨ıdes et polymatro¨ıdes etiquet´ es´ definis´ sur un ensemble fixe, et leurs equivalents´ pas etiquet´ es,´ les Z-modules des invariants valuatifs. Nous fournissons des bases des ces modules et leurs modules duels, engendres´ par fonctions characteristiques´ des poly- topes, et des formules explicites donnants leurs rangs. Nos resultats´ confirment une conjecture du premier auteur, que G soit universel pour les invariants valuatifs. Keywords: polymatroids, polymatroid polytopes, decompositions, valuations 1 Introduction Matroids were introduced by Whitney in 1935 (see [22]) as a combinatorial abstraction of linear depen- dence of vectors in a vector space.
    [Show full text]
  • Nick Early.Pdf
    From matroid subdivisions of hypersimplices to generalized Feynman diagrams Nick Early Perimeter Institute for Theoretical Physics March 4, 2020 Nick Early Matroid subdivisions and generalized Feynman diagrams Preview slide {1, 4} {1, 3} {3, 4} {1, 4} {1, 3} {1, 2} {3, 4} {2, 4} {2, 3} {1, 2} {2, 4} {2, 3} Blades on the Universal Octahedron. Nick Early Matroid subdivisions and generalized Feynman diagrams The unreasonable effectiveness of mathematics. Background. I'll tell two parallel stories, both starting in 2012-2013, which ultimately joined forces in 2018/2019: my (math) Ph.D. thesis on symmetries and invariants for subdivisions of hypersimplices, and the CHY method for computing biadjoint scattering amplitudes. Nick Early Matroid subdivisions and generalized Feynman diagrams (k) introduce the generalization m (In; In) with k ≥ 3. Objective (2) introduce a new mathematical object, the matroidal blade arrangement, on the vertices of kth hypersimplex ∆k;n. Objective (3): show how these arrangements select a natural set of planar functions on the kinematic space for all n and k. These define a basis, the planar basis. Remark: These specialize for k = 2 to the planar basis introduced by [CHY2013] and denoted Xij in [ABHY2017]. Objectives Objective (0): interpret the biadjoint scalar scattering (k=2) amplitude m (In; In) in terms of matroid subdivisions of a convex polytope known as the (second) hypersimplex, ∆2;n. Objective (1): Nick Early Matroid subdivisions and generalized Feynman diagrams show how these arrangements select a natural set of planar functions on the kinematic space for all n and k. These define a basis, the planar basis.
    [Show full text]
  • Basis Graphs of Even Delta-Matroids
    ARTICLE IN PRESS YJCTB:2408 JID:YJCTB AID:2408 /FLA [m1+; v 1.60; Prn:7/06/2006; 16:40] P.1 (1-18) Journal of Combinatorial Theory, Series B ••• (••••) •••–••• www.elsevier.com/locate/jctb Basis graphs of even Delta-matroids Victor Chepoi Laboratoire d’Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, Université de la Méditerranée, F-13288 Marseille cedex 9, France Received 16 February 2000 Abstract A -matroid is a collection B of subsets of a finite set I, called bases, not necessarily equicardinal, satisfying the symmetric exchange property: For A,B ∈ B and i ∈ AB, there exists j ∈ BAsuch that (A {i, j}) ∈ B.A-matroid whose bases all have the same cardinality modulo 2 is called an even -matroid.Thebasis graph G = G(B) of an even -matroid B is the graph whose vertices are the bases of B and edges are the pairs A,B of bases differing by a single exchange (i.e., |AB|=2). In this note, we present a characterization of basis graphs of even -matroids, extending the description of basis graphs of ordinary matroids given by S. Maurer in 1973: Theorem. A graph G = (V, E) is a basis graph of an even -matroid if and only if it satisfies the following conditions: (a) if x1x2x3x4 is a square and b ∈ V ,thend(b,x1) + d(b,x3) = d(b,x2) + d(b,x4); (b) each 2-interval of G contains a square and is an induced subgraph of the 4-octahedron; (c) the neighborhoods of vertices induce line graphs, or, equivalently, the neighborhoods of vertices do not contain induced 5- and 6-wheels.
    [Show full text]
  • Arxiv:Math/0411260V1
    MATROID POLYTOPES, NESTED SETS AND BERGMAN FANS EVA MARIA FEICHTNER AND BERND STURMFELS Abstract. The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained intro- duction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences. 1. Introduction Let V be an r-dimensional vector subspace of the n-dimensional vector space Cn over the field C of complex numbers. The amoeba of V is the set of all vectors of the form n log|v1|, log|v2|,..., log|vn| ∈ R , where (v1, . , vn) runs over all vectors in V whose coordinates are non-zero. The asymp- totic behavior of the amoeba is given by an r-dimensional polyhedral fan in Rn. This fan was called the logarithmic limit set of V in George Bergman’s seminal paper [3]. We use the term Bergman fan for this object. The study of such polyhedral spaces is now an active area of research, known as tropical geometry [14, 18, 19, 21].
    [Show full text]