Annals of Combinatorics 1 (1997) 159-172


Pfaffian Structures and Critical Problems in Finite Symplectic Spaces

Joseph P.S. Kung

Department of Mathematics, University of North Texas, Denton, Texas 76203, USA
kung@unt.edu

Received March 18, 1997

AMS Subject Classification: 05B35; 05B25, 05C70, 05C35, 05E15, 15A15

Abstract. Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a set S of non-zero vectors in a non-singular symplectic space V of dimension 2m, find its symplectic critical exponent, that is, the minimum of the set {m - dim(U): US= Ø}, where U ranges over all the (totally) isotropic subspaces disjoint from S. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the set S by Möbius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid on S given by linear dependence. This formula implies that the symplectic critical exponent of S depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic space V.

Keywords: Matroids, Pfaffians, symplectic space, basis exchange axioms, critical problem, isotropic subspace, Möbius inversion


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