The Tutte Dichromate and Whitney Homology of Matroids

We consider a specialization YM (q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M . We show that the coefficients of YM (1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of M . In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid. 0. Introduction. In 1954, Tutte [30] introduced the dichromate of a (finite) graph, which has since become known as the Tutte polynomial. In the four decades since then this has provided a profound link between combinatorics and other branches of mathematics as diverse as statistical mechanics [3, 5, 17, 19, 20], low-dimensional topology [17, 18, 20], and the theory of Grothendieck rings [9, 10, 29]. Indeed, in 1947 Tutte [29] showed that the “Tutte-Grothendieck ring” K0(G) of a suitably defined category G of graphs is Z[x, y]; the class TG(x, y) in K0(G) of a graph G is its Tutte polynomial. This construction was axiomatized for “bidecomposition categories” and applied to a category M of matroids by Brylawski [9, 10], with the result that K0(M) = Z[x, y] as well. (It is interesting to compare Brylawski’s axiomatization with the usual hypotheses of algebraic K-theory; see, e.g. Chapter 5 of Silvester [27].) Crapo’s generalization [14] of Tutte polynomials to matroids rests on this foundation. The fact that TM(x, y) is the “universal Tutte-Grothendieck invariant” of the matroid M is just a restatement of the fact that it is the class of M in K0(M). This categorical perspective overlooks the question of what combinatorial information about a matroid is encoded in its Tutte polynomial. Much of the inquiry into 1991 Mathematics Subject Classification. 05B35, 06A08.