The Combinatorial Laplacian of the Tutte Complex

Let M be an ordered matroid and C••(M) be an exterior algebra over its underlying set E, graded by both corank and nullity. Then C•0(M) is the simplicial chain complex of IN(M), the simplicial complex whose simplices are indexed by the independent sets of the matroid. Dually, C0•(M) is the cochain complex of IN(M*). We give a combinatorial description of a basis of eigenvectors for the combinatorial Laplacian of a family of boundary maps on the double complex, extending work by W. Kook, V. Reiner, and D. Stanton [2000, J. Amer. Math. Soc.13, 129–148] on IN(M). The eigenvalues are enumerated by a weighted version of the Tutte polynomial, using an identity of G. Etienne and M. Las Vergnas [1998, Discrete Math.179, 111–119]. As an application, we prove a duality theorem for the cohomology of Orlik–Solomon algebras.