# Specht Polytopes and Specht Matroids

## TL;DR

This paper introduces Specht matroids and polytopes to capture relations in Specht modules, extends the concept to Kronecker coefficients, and provides computational tools, offering new geometric perspectives in representation theory.

## Contribution

It defines Specht and Kronecker matroids and polytopes, introduces the concept of matroidification, and supplies computational code for these structures.

## Key findings

- Symmetric group acts transitively on Specht polytope vertices.
- Specht polytope's ambient space is irreducible under group action.
- Provides elementary construction of Specht modules and computational tools.

## 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, 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.