Quadratic algebras and line-closed matroids
by
Michael Falk
Northern Arizona University

Let G be a matroid and A(G) the associated Orlik-Solomon algebra. Recall that A is quadratic if and only if the relation ideal is generated by elements of degree two. A set S of points is line-closed in G if, for every x, y in S, the entire line spanned by x and y is contained in S. The matroid G is line-closed if any line-closed set is closed.

In this talk we describe work in progress toward a proof of the following conjecture: A(G) is quadratic if and only if G is line-closed. The proof is more interesting than the conjecture. Fix a linear order of the points of G. We construct a natural generalization of the well-known nbc basis of A(G), which contains the usual basis, and coincides with it if and only if G is line-closed. We prove that these 2-nbc bases are linearly independent in the quadratic closure of A(G), yielding one of the implications in the conjecture.

The number of 2-nbc bases changes when the linear order is changed, a reflection of the fact that line-closure does not satisfy the Steinitz exchange axiom. We give some indications that for some special linear orders, the 2-nbc bases give bases for the quadratic closure of A(G), yielding a proof of the conjecture and a combinatorial calculation of the ``global invariant" \phi₃(A) introduced in our earlier work. We also describe how this idea can be extended to give a sequence of combinatorial/algebraic/topological invariants of matroids, OS algebras, or hyperplane arrangements.

Date received: May 26, 1999