Valuations and the Hopf Monoid of Generalized Permutahedra

Ardila, Federico; García, Mario Sánchez

doi:10.1093/imrn/rnab355

Cited by 14 publications

(26 citation statements)

References 35 publications

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“…Many invariants of matroids satisfy valuativity, including the Tutte polynomial and its specializations [AFR10,DF10]. For a more comprehensive list see [AS20], and see [ESS21] for a study of valuativity for Coxeter matroids. We show that a wide range of classes associated to the tautological K-classes of matroids are also valuative.…”

Section: Base Polytope Propertiesmentioning

confidence: 99%

Tautological classes of matroids

Berget¹,

Eur²,

Spink³

et al. 2021

Preprint

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We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory.

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Section: Base Polytope Propertiesmentioning

confidence: 99%

Tautological classes of matroids

Berget¹,

Eur²,

Spink³

et al. 2021

Preprint

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“…What can be said about 0/1 extended generalized permutohedra with the same indicator complex (whether or not it is a matroid complex)? (15) Ardila and Sanchez [AS20] recently studied valuations of generalized permutohedra by passing to a quotient of GP `by inclusion/exclusion relations, as in Mc-Mullen's polytope algebra [McM89]. They showed that this quotient is isomorphic to a Hopf monoid of weighted ordered partitions.…”

Section: Proofmentioning

confidence: 99%

“…The Hopf algebra of matroids was introduced by Crapo and Schmitt [CS05a,CS05b,CS05c]. The corresponding Hopf monoid was described by Aguiar and Mahajan [AM10, §13.8.2] and has attracted recent interest; see, e.g., [San20,Sup20,AS20,Bas20]. There are many definitions of a matroid (see, e.g., [Oxl11, Section 1]), but for our purposes the most convenient definition is that a matroid is a simplicial complex Γ on vertex set I such that the induced subcomplex Γ|S " tσ P Γ : σ Ď Su is pure for every S Ď I (i.e., every facet of Γ|S has the same size).…”

Section: Introductionmentioning

confidence: 99%

Hopf monoids of ordered simplicial complexes

Castillo,

Martin,

Samper

2020

Preprint

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We study pure ordered simplicial complexes (i.e., simplicial complexes with a linear order on their ground sets) from the Hopf-theoretic point of view. We define a Hopf class to be a family of pure ordered simplicial complexes that give rise to a Hopf monoid under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class allows us to give a systematic study of simplicial complexes related to matroids, including shifted complexes, broken-circuit complexes, and unbounded matroids (which arise from unbounded generalized permutohedra with 0/1 coordinates).We compute the antipodes in two cases: facet-initial complexes (a much larger class than shifted complexes) and unbounded ordered matroids. In the latter case, we embed the Hopf monoid of ordered matroids into the Hopf monoid of ordered generalized permutohedra, enabling us to compute the antipode using the topological method of Aguiar and Ardila. The calculation is complicated by the appearance of certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control certain coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellationfree.

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“…Many matroid invariants, including the Tutte polynomial, the Kazhdan-Lusztig polynomial, the motivic zeta function, the Chern-Schwartz-MacPherson cycle, and the volume polynomial of the Chow ring, turn out to be valuative. See [AFR10,AS22,Ard22] for extensive lists and history of the study of valuative matroid invariants.…”

Section: Introductionmentioning

confidence: 99%

Stellahedral geometry of matroids

Eur¹,

Huh²,

Larson³

2022

Preprint

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We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety, and show that valuative, homological, and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shaprio-Smirnov-Vaintrob on Postnikov-Shaprio algebras, and calculate the Chern-Schwartz-MacPherson classes of matroid Schubert cells. The central construction is the "augmented tautological classes of matroids," modeled after certain vector bundles on the stellahedral toric variety. CONTENTS 1. Introduction 1 2. Torus-equivariant geometry preliminaries 3. Stellahedral varieties 4. Augmented tautological bundles and classes 5. Augmented wonderful varieties and Bergman classes 6. Exceptional isomorphisms 7. Valuative group, homology, and the intersection pairing 8. Numerical properties 9. Chern-Schwartz-MacPherson classes Appendix A. Polytope algebras and K-rings of toric varieties References

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Valuations and the Hopf Monoid of Generalized Permutahedra

Cited by 14 publications

References 35 publications

Tautological classes of matroids

Tautological classes of matroids

Hopf monoids of ordered simplicial complexes

Stellahedral geometry of matroids

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