A Matroid Invariant via the K-Theory of the Grassmannian

Let 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)[OTx], gx behaves nicely under the standard constructions of matroid theory. Specifically, gx1⊕x2(t) = gx1(t)gx2(t), gx1+2 x2(t) = gx1(t)gx2(t)/t, gx(t) = gx⊥(t) and gx is unaltered by series and parallel extensions. Furthermore, the coefficients of gx are nonnegative. The existence of this map implies bounds on (essentially equivalently) the complexity of Kapranov’s Lie complexes [13], Hacking, Keel and Tevelev’s very stable pairs [11] and the author’s tropical linear spaces when they are realizable in characteristic zero [25]. 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. 1 Motivation and Introduction Let K = ⋃∞ n=1 C((t 1/n)), the field of Puiseux series, and let v : K∗ → Q be the map which assigns to a power series its order of vanishing; in other words, if x = ∑ i≥M ait i/N with aM 6= 0 then v(x) = M/N . Suppose that we have a K-valued point of the Grassmannian G(d, n) none of whose Plücker coordinates pI(x) are zero. In [26], Sturmfels and I attempted to determine the possible (n d ) -tuples of rational numbers v(pI(x)). The pI(x) obey the Plücker relations: pSijpSkl − pSikpSjl + pSilpSjk = 0 for any S ∈ ( [n] d−2 ) 1 and i < j < k < l in [n] \ S. As a consequence, we deduce that Through out this paper, we use the following combinatorial notations: [n] = {1, 2, . . . , n} and, for any set S, ( S d ) is the set of d element subsets of S. We will sometimes, as in the equation above, use the shorthand Si for S ∪ {i}.