Lagrangian combinatorics of matroids

Ardila, Federico; Denham, Graham; Huh, June

doi:10.5802/alco.263

Ardila, Federico ¹; Denham, Graham ²; Huh, June ³

¹ San Francisco State University Dept. of Mathematics 1600 Holloway Ave. San Francisco, CA 94110 USA Universidad de Los Andes Depto. de Matemáticas Cra. 1 #18a-12 Bogotá Colombia
² University of Western Ontario Dept. of Mathematics Middlesex College London, Ontario Canada N6A 5B7
³ Princeton University Dept. of Mathematics Fine Hall 304 Washington Rd. Princeton, NJ 08544 USA

Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 387-411.

Abstract

The Lagrangian geometry of matroids was introduced in [2] through the construction of the conormal fan of a matroid $M$ . We used the conormal fan to give a Lagrangian-geometric interpretation of the $h$ -vector of the broken circuit complex of $M$ : its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $γ$ and $δ$ on the conormal fan of $M$ . By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski’s conjecture that this $h$ -vector is a log-concave sequence.

This sequel explores the Lagrangian combinatorics of matroids, further developing the combinatorics of biflats and biflags of a matroid, and relating them to the theory of basis activities developed by Tutte, Crapo, and Las Vergnas. Our main result is a combinatorial realization of the intersection-theoretic computation above: we write the $k$ -th mixed intersection of $γ$ and $δ$ explicitly as a sum of biflags corresponding to the nbc bases of internal activity $k + 1$ .

Received: 2021-12-12
Revised: 2022-01-31
Accepted: 2022-08-10
Published online: 2023-05-03

DOI: 10.5802/alco.263

Classification: 05B35, 05E45, 14T15, 52B40
Keywords: basis activity, conormal fan, $h$-vector, matroid

Author's affiliations:

Ardila, Federico ¹; Denham, Graham ²; Huh, June ³

¹ San Francisco State University Dept. of Mathematics 1600 Holloway Ave. San Francisco, CA 94110 USA Universidad de Los Andes Depto. de Matemáticas Cra. 1 #18a-12 Bogotá Colombia
² University of Western Ontario Dept. of Mathematics Middlesex College London, Ontario Canada N6A 5B7
³ Princeton University Dept. of Mathematics Fine Hall 304 Washington Rd. Princeton, NJ 08544 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Ardila, Federico; Denham, Graham; Huh, June. Lagrangian combinatorics of matroids. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 387-411. doi : 10.5802/alco.263. https://alco.centre-mersenne.org/articles/10.5802/alco.263/

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