A poset Φn whose maximal chains are in bijection with the n × n alternating sign matrices

For an integer n≥1, we display a poset Φn whose maximal chains are in bijection with the n×n alternating sign matrices. The Hasse diagram Φˆn is obtained from the n-cube by adding some edges. We show that the dihedral group D2n acts on Φˆn as a group of automorphisms.