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 \bar{Tx} be the closure of the T-orbit through x. Then the class of the structure sheaf of \bar{Tx} in the K-theory of G(d,n) depends only on which Pl\"ucker 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 the K-theory of G(d,n) to Z[t]. Letting g_x(t) denote the image of (-1)^{n-dim Tx} [ O_{\bar{Tx}}], g_x behaves nicely under the standard constructions of matroid theory. Specifically, g_{x_1 \oplus x_2}(t)=g_{x_1}(t) g_{x_2}(t), g_{x_1 +_2 x_2}(t)=g_{x_1}(t) g_{x_2}(t)/t, g_x(t) = g_{x^{\perp}}(t) and g_x is unaltered by series and parallel extensions. Furthermore, the coefficients of g_x are nonnegative. The existence of this map implies bounds on (essentially equivalently) the complexity of Kapranov's Lie complexes, Hacking, Keel and Tevelev's very stable pairs and the author's tropical linear spaces when they are realizable in characteristic zero. 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 and a tropical linear space realizable in characteristic zero can have at most this many i-dimensional bounded faces.