Combinatorial Algebraic Geometry pp 201-228 | Cite as

# 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.

## MSC 2010 codes:

05E10## Notes

### Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The authors wish to thank Bernd Sturmfels, Diane Maclagan, Gregory G. Smith for their leadership and encouragement, and all the participants of the Combinatorial Algebraic Geometry thematic program at the Fields Institute, at which this work was conceived. They also thank the Fields Institute and the Clay Mathematics Institute for hospitality and support. Finally, thanks to Bernd Sturmfels also for suggesting the term “matroidification” and to anonymous referees for their helpful suggestions.

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