Gr\"obner fans of Specht ideals

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 $i$.