Gr\"obner fans of Specht ideals

TL;DR

This paper explicitly describes the Gr"obner fan and state polytope of Specht ideals, revealing that the state polytope is always a generalized permutohedron and characterizing when it is a permutohedron.

Contribution

It provides an explicit description of the Gr"obner fan and state polytope of Specht ideals, linking their geometric structure to generalized permutohedra.

Findings

The state polytope of a Specht ideal is always a generalized permutohedron.

The state polytope is a permutohedron if and only if certain parts of the partition are equal.

Explicit descriptions of the Gr"obner fan for Specht ideals are provided.

Abstract

In this paper, we give the Gr\"obner fan and the state polytope of a Specht ideal $I_\lambda$ explicitly. In particular, we show that the state polytope of $I_\lambda$ for a partition $\lambda=(\lambda_1, \ldots, \lambda_m)$ is always a generalized permutohedron, and it is a (usual) permutohedron if and only if $\lambda_{i-1}=\lambda_i>0$ for some $i$.