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, Tome 6 (2023) no. 2, pp. 387-411

Résumé

The Lagrangian geometry of matroids was introduced in [2] through the construction of the conormal fan of a matroid $\mathrm{M}$. We used the conormal fan to give a Lagrangian-geometric interpretation of the $h$-vector of the broken circuit complex of $\mathrm{M}$: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $\gamma$ and $\delta$ on the conormal fan of $\mathrm{M}$. 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 $\gamma$ and $\delta$ explicitly as a sum of biflags corresponding to the nbc bases of internal activity $k+1$.

Détail

Reçu le : 2021-12-12
Révisé le : 2022-01-31
Accepté le : 2022-08-10
Publié le : 2023-05-03

DOI : 10.5802/alco.263

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits