Stellahedral geometry of matroids

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-conca...

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Bibliographic Details
Published inForum of mathematics. Pi Vol. 11
Main Authors Eur, Christopher, Huh, June, Larson, Matt
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 09.10.2023
Subjects
Concavity
Discrete Mathematics
Homology
Polynomials
Rings (mathematics)
14N20
52C35
05B25
14M25
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ISSN2050-5086
2050-5086
DOI10.1017/fmp.2023.24

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Summary: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 Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro 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 toric vector bundles on the stellahedral toric variety.
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ISSN:2050-5086
2050-5086
DOI:10.1017/fmp.2023.24