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, 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.The irreducible representations of the symmetric group S n were worked out by Young and Specht in the early 20th century, and they remain omnipresent in algebraic combinatorics. The symmetric group S n has a unique irreducible representation for each partition of n. For example, S 4 has exactly five irreducible representations corresponding to the partitions (4) (

The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group

“…The convex hull of the set of normals {n α } α∈Pn(λ ′ ) is an important polytope (in another form it appeared in [11] under the name of "Specht polytope"). For instance, in the case λ = (n − 1, 1) of the braid arrangement, it is the root polytope of type A n .…”

“…For a background on hyperplane arrangements and related objects, we 1 When this article was under preparation, we have come upon the paper [11] in which closely related objects are considered. However, the approach in [11] is completely different, in that, first, no hyperplane arrangements are considered, and, second, the definitions are given just in a purely combinatorial form, while our constructions are systematically defined in invariant representation-theoretic terms.…”