# ALGEBRAIC COMBINATORICS

Representability of orthogonal matroids over partial fields
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1301-1311.

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 $\mathrm{Supp}\left(p\right):=\left\{J\in \left(\genfrac{}{}{0pt}{}{E}{r}\right)\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}{p}_{J}\ne 0\right\}$ 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.

Accepted:
Published online:
DOI: 10.5802/alco.301
Classification: 05B35, 12K99
Keywords: matroid, Grassmannian, orthogonal matroid
Baker, Matthew 1; Jin, Tong 1

1 Georgia Institute of Technology School of Mathematics 686 Cherry Street Atlanta GA 30332-0160 (USA)
License: CC-BY 4.0
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Baker, Matthew; Jin, Tong. Representability of orthogonal matroids over partial fields. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1301-1311. doi : 10.5802/alco.301. https://alco.centre-mersenne.org/articles/10.5802/alco.301/

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