Stellahedral geometry of matroids

Eur, Christopher; Huh, June; Larson, Matt

doi:10.48550/arxiv.2207.10605

Cited by 3 publications

(3 citation statements)

References 33 publications

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“…Remark 3 Our definition of Schubert matroids is not closed under isomorphism since it depends on an ordering of the ground set. Definitions that do not depend on the ordering can be found in [24,Definition 7.5] and [27,Definition 2.20] These and isomorphic matroids are also known as "generalized Catalan matroids", "shifted matroids", "nested matroids" and "freedom matroids" (see the discussion in [5, Section 4].…”

Section: Matroids Positroids and The (Real) Grassmannianmentioning

confidence: 99%

Lattice Path Matroids and Quotients

Benedetti-Velásquez,

Knauer

2024

Combinatorica

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We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

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Section: Matroids Positroids and The (Real) Grassmannianmentioning

confidence: 99%

Lattice Path Matroids and Quotients

Benedetti-Velásquez,

Knauer

2024

Combinatorica

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“…We show, in Corollary 5.9, that these lattice polytopes are anti-blocking versions of certain permutohedra. We also note (see Remark 3.19) that for any n ≥ m, P(m, n) is combinatorially equivalent to the m-stellohedron which, for example, has been studied in [12,Section 10.4] and has appeared recently in connection with matroid theory [6].…”

Section: Introductionmentioning

confidence: 99%

Partial permutohedra

Behrend¹,

Castillo²,

Chavez³

et al. 2022

Preprint

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Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker [arXiv:2012.09901]. For positive integers m and n, the partial permutohedron P(m, n) is the convex hull of all vectors in {0, 1, . . . , n} m with distinct nonzero entries. We study the face lattice, volume and Ehrhart polynomial of P(m, n), and our methods and results include the following. For any m and n, we obtain a bijection between the nonempty faces of P(m, n) and certain chains of subsets of {1, . . . , m}, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the h-polynomial of P(m, n). For any m and n with n ≥ m − 1, we use a pyramidal subdivision of P(m, n) to obtain a recursive formula for the volume of P(m, n). We also use a sculpting process (in which P(m, n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of P(m, n) with arbitrary m and fixed n ≤ 3, the volume of P(m, 4) with arbitrary m, and the Ehrhart polynomial of P(m, n) with fixed m ≤ 4 and arbitrary n ≥ m − 1.

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“…To every matroid M one may associate its base polytope (M), carrying all the information of the matroid. Not only this polytope plays prominent roles in combinatorial otpimization [Sch03], but also is of fundamental importance in tropical geometry [MS15,Jos21], the theory of valuations [DF10,AS23], combinatorial Hodge theory, and the study of matroid invariants [BEST23,EHL22,FS22].…”

mentioning

confidence: 99%

Matroid relaxations and Kazhdan–Lusztig non-degeneracy

Ferroni¹,

Vecchi²

2022

Algebraic Combinatorics

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In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan-Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan-Lusztig, the inverse Kazhdan-Lusztig and the Z-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are nondegenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan-Lusztig, inverse Kazhdan-Lusztig and Z-polynomial of all sparse paving matroids.Conjecture 1.1 ([13, Conjecture 3.2]). The Kazhdan-Lusztig polynomial of a matroid is real-rooted.

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Stellahedral geometry of matroids

Cited by 3 publications

References 33 publications

Lattice Path Matroids and Quotients

Lattice Path Matroids and Quotients

Partial permutohedra

Matroid relaxations and Kazhdan–Lusztig non-degeneracy

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