A matroid invariant via the K-theory of the Grassmannian

AbstractLet G(d,n) denote the Grassmannian of d-planes in Cn 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 Tx¯ be the closure of the T-orbit through x. Then the class of the structure sheaf of Tx¯ 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 gx(t) denote the image of (−1)n−dimTx[OTx¯], gx 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