Specht polytopes and specht matroids

Abstract

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: 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 call this process of upgrading from numbers to matroids and polytopes “matroidification”. In the course of describing these objects, we also give an elementary account of the construction of Specht modules. Finally, we provide code to compute with Specht matroids and their Chow rings.