Specht Polytopes and Specht Matroids

The generators of the classical Specht module satisfy intricate relations. We introduce the Specht matroid, which keeps track of these relations, and the Specht polytope, which also keeps track of convexity relations. We establish basic facts about the Specht polytope, for example, that the symmetric group acts transitively on its vertices and irreducibly on its ambient real vector space. A similar construction builds a matroid and polytope for a tensor product of Specht modules, giving "Kronecker matroids" and "Kronecker polytopes" instead of the usual Kronecker coefficients. We dub this process of upgrading numbers to matroids and polytopes "matroidification," giving two more examples. In the course of describing these objects, we also give an elementary account of the construction of Specht modules different from the standard one. Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure

A module-theoretic approach to matroids

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA

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