A matroid invariant via the K-theory of the Grassmannian

Abstract

Let G ( d , n ) denote the Grassmannian of d-planes in C n and let T be the torus ( C ∗ ) n / diag ( C ∗ ) which acts on G ( d , n ) . Let x be a point of G ( d , n ) and let T x ¯ be the closure of the T-orbit through x. Then the class of the structure sheaf of T x ¯ in the K-theory of G ( d , n ) depends only on which Plücker coordinates of x are nonzero – combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K ○ ( G ( d , n ) ) to Z [ t ] . Letting g x ( t ) denote the image of ( − 1 ) n − dim T x [ O T x ¯ ] , g x behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (2) (1993) 29–110], Hacking, Keel and Tevelev's very stable pairs [P. Hacking, S. Keel, E. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom. 15 (2006) 657–680] and the author's tropical linear spaces when they are realizable in characteristic zero [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558]. Namely, in characteristic zero, a Lie complex or the underlying ( d − 1 ) -dimensional scheme of a very stable pair can have at most ( n − i − 1 ) ! ( d − i ) ! ( n − d − i ) ! ( i − 1 ) ! strata of dimensions n − i and d − i , respectively. This prove the author's f-vector conjecture, from [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558], in the case of a tropical linear space realizable in characteristic 0.