ISSN:
1572-9192
Keywords:
symplectic matroid
;
Coxeter matroid
;
totally isotropic subspace
;
symmetric matroid
;
2-matroid
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i ≤ n, and which has the additional property that for any linear ordering ≺ of J such that i ≺ j implies j* ≺ i* and i ≺ j* implies j ≺ i* for all i, j ≤ n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for 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 subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008610715466
Permalink
