A poset $\Phi_n$ whose maximal chains are in bijection with the $n \times n$ alternating sign matrices

TL;DR

This paper constructs a poset whose maximal chains correspond to $n imes n$ alternating sign matrices, revealing symmetries and automorphisms related to the dihedral group.

Contribution

It introduces a new poset $\Phi_n$ with a bijection to alternating sign matrices and explores its symmetry properties under the dihedral group.

Findings

Maximal chains in $\Phi_n$ correspond to $n imes n$ alternating sign matrices.

The Hasse diagram is derived from the $n$-cube with added edges.

The dihedral group $D_{2n}$ acts as automorphisms on the diagram.

Abstract

For an integer $n\geq 1$, we display a poset $\Phi_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat \Phi_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat \Phi_n$ as a group of automorphisms.