Matroid base polytope decomposition

Abstract

Let P (M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P (M) is a decomposition of the formis also a matroid base polytope for some matroid M i , and for each 1In this paper, we investigate hyperplane splits, that is, polytope decompositions when t = 2. We give sufficient conditions for M so P (M) has a hyperplane split and characterize when P (M 1 ⊕ M 2 ) has a hyperplane split where M 1 ⊕ M 2 denote the direct sum of matroids M 1 and M 2 . We also prove that P (M) has not a hyperplane split if M is binary. Finally, we show that P (M)has not a decomposition if its 1-skeleton is the hypercube.

Matroid base polytope decomposition II: Sequences of hyperplane splits

“…This paper is a continuation of the paper [3] by the two present authors. For general background in matroid theory we refer the reader to [13,16] where e i is the i th standard basis vector in R n .…”

“…Matroid base polytope decomposition were introduced by Lafforgue [10,11] and have appeared in many different contexts : quasisymmetric functions [1,2,4,12], compactification of the moduli space of hyperplane arrangements [6,8], tropical linear spaces [14,15], etc. In [3], we have studied the existence (and nonexistence) of such decompositions. Among other results, we presented sufficient conditions on a matroid M so P (M ) admits a hyperplane split.…”

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

“…st(U ) = (1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0) P = {4, 5, 9, 10, 13} st(P ) = (0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1) L = {5, 8, 10, 12, 13} st(L) = (0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1) Figure 1. Left: Lattice paths U and L from (0, 0) to (8,5) and a path P staying between U and L in the diagram of M [U, L]. Right: Representations of U , L, and P as subsets of {1, .…”

“…The class of lattice path matroids was first introduced by Bonin, de Mier, and Noy [5]. Many different aspects of lattice path matroids have been studied: excluded minor results [4], algebraic geometry notions [12,23,24], the Tutte polynomial [6,18,19], the associated basis polytope in connection with the combinatorics of Bergman complexes [12], its facial structure [1,2], specific decompositions in relation with Lafforgue's work [8] as well as the related cut-set expansion conjecture [9].…”

A Tutte polynomial inequality for lattice path matroids

“…In this section we address the class of lattice path matroids first introduced by Bonin, de Mier, and Noy [4]. Many different aspects of lattice path matroids have been studied: excluded minor results [2], algebraic geometry notions [9,17,18], complexity of computing the Tutte polynomial [5,15], and results around the matroid base polytope [6].…”