ALGEBRAIC COMBINATORICS

no. 2
Lagrangian combinatorics of matroids
Ardila, Federico1; Denham, Graham2; Huh, June3
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.

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.263
Classification: 05B35, 05E45, 14T15, 52B40
Keywords: basis activity, conormal fan, $h$-vector, matroid
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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