ALGEBRAIC COMBINATORICS

no. 5
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, Volume 6 (2023) no. 5, pp. 1301-1311.

Let r≤n be nonnegative integers, and let N=n r-1. For a matroid M of rank r on the finite set E=[n] 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=(p J )∈P N (k) with support M (meaning that Supp(p):={J∈E r|p J ≠0} of p is the set of bases of M) satisfying the Grassmann-Plücker equations; and (c) there is a point p=(p J )∈P N (k) 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.

Received:
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
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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