Representability of orthogonal matroids over partial fields

Baker, Matthew¹ ; Jin, Tong¹

¹ Georgia Institute of Technology School of Mathematics 686 Cherry Street Atlanta GA 30332-0160 (USA)

Algebraic Combinatorics, Tome 6 (2023) no. 5, pp. 1301-1311

Résumé

Let $r\le n$ be nonnegative integers, and let $N=\left(\genfrac{}{}{0pt}{}{n}{r}\right)-1$. For a matroid $M$ of rank $r$ on the finite set $E=\left[n\right]$ and a partial field $k$ in the sense of Semple–Whittle, it is known that the following are equivalent: (a) $M$ is representable over $k$; (b) there is a point $p=\left(p_J\right)\in {\mathbf{P}}^N\left(k\right)$ with support $M$ (meaning that $\mathrm{Supp}\left(p\right):=\{J\in \left(\genfrac{}{}{0pt}{}{E}{r}\right)|p_J\ne 0\}$ of $p$ is the set of bases of $M$) satisfying the Grassmann-Plücker equations; and (c) there is a point $p=\left(p_J\right)\in {\mathbf{P}}^N\left(k\right)$ with support $M$ satisfying just the 3-term Grassmann-Plücker equations. Moreover, by a theorem of P. Nelson, almost all matroids (meaning asymptotically 100%) are not representable over any partial field. We prove analogues of these facts for Lagrangian orthogonal matroids in the sense of Gelfand–Serganova, which are equivalent to even Delta-matroids in the sense of Bouchet.

Détail

Reçu le : 2022-09-23
Accepté le : 2023-01-31
Publié le : 2023-11-07

DOI : 10.5802/alco.301

Classification : 05B35, 12K99
Keywords: matroid, Grassmannian, orthogonal matroid

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits