Symplectic Matroids

• Authors: Alexandre V. Borovik;Israel Gelfand;Neil White
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 1998

A symplectic matroid is a collection {\cal B} of k-elementsubsets of J = {1, 2, …, n, 1*,2*, … n*}, eachof which containsnot both of i and i* for every i ≤ n, and which has the additionalproperty that for any linear ordering ≺ of J such that i ≺ j implies j* ≺ i* and i ≺ j* implies j ≺i* forall i, j ≤ n, {\cal B}has a member which dominates element-wise every othermember of {\cal B}. Symplectic matroids are a special case of Coxetermatroids, namely the case where the Coxeter group is the hyperoctahedralgroup, the group of symmetries of the n-cube. In this paper we developthe basic properties of symplectic matroids in a largely self-containedand elementary fashion. Many of these results are analogous to resultsfor ordinary matroids (which are Coxeter matroids for the symmetric group),yet most are not generalizable to arbitrary Coxeter matroids. For example,representable symplectic matroids arise from totally isotropic subspacesof a symplectic space very similarly to the way in which representableordinary matroids arise from a subspace of a vector space. We also examineLagrangian matroids, which are the special case of symplectic matroidswhere k = n, and which are equivalent to Bouchet‘s symmetric matroids or 2-matroids.