Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope

The polytope $AS{M}_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $AS{M}_n$ corresponds to an elementary flow grid defined by Striker, and each elementary flow grid determines a doubly directed graph defined by Brualdi and Dahl. We show that a face of $AS{M}_n$ is symmetric if and only if its doubly directed graph has all vertices of even degree. We show that every face of $AS{M}_n$ is a 2-level polytope. We show that a $d$-dimensional face of $AS{M}_n$ has at most $2^d$ vertices and $4(d-1)$ facets, for $d\ge 2$. We show that a $d$-dimensional face of $AS{M}_n$ satisfies $vf\le d{2}^{d+1}$, where $v$ and $f$ are the numbers of vertices and edges of the face. If the doubly directed graph of a $d$-dimensional face is 2-connected, then $v\le 2^{d-1}+2$. We describe the facets of a face and a basis for the subspace parallel to a face in terms of the elementary flow grid of the face. We prove that no face of $AS{M}_n$ has the combinatorial type of the Birkhoff polytope $B_3$. We list the combinatorial types of faces of $AS{M}_n$ that have dimension 4 or less.