A free subalgebra of the algebra of matroids

This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and a single loop in the dual of this Hopf algebra is free.Comment: 19 pages, 3 figures. Accepted for publication in the European Journal of Combinatorics. This version incorporates a few minor corrections suggested by the publisher

Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal G$ introduced by the first author, are valuative. In this paper we construct the $\Z$-modules of all $\Z$-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the $\Z$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal G$ is universal for valuative invariants.Comment: 54 pp, 9 figs. Mostly minor changes; Cor 10.5 and formula for products of $u$s corrected; Prop 7.2 is new. To appear in Advances in Mathematic

Algebras related to matroids represented in characteristic zero

Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the structure of these algebras. In return, the numerical properties of the Hilbert function of A yield some information about the Tutte polynomial of the corresponding matroid. Isomorphism classes of these algebras correspond to equivalence classes of hyperplane arrangements under the action of the general linear group.Comment: 11 pages AMS-LaTe

Proto-exact categories of matroids, Hall algebras, and K-theory

This paper examines the category $\mathbf{Mat}_{\bullet}$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that $\mathbf{Mat}_{\bullet}$ has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory $K_* (\mathbf{Mat}_{\bullet})$ of $\mathbf{Mat}_{\bullet}$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $$\pi^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet})$$ from the stable homotopy groups of spheres for all $n$. Finally, we show that the Hall algebra of $\mathbf{Mat}_{\bullet}$ is a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.Comment: 29 page