Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope

TL;DR

This work analyzes the geometry and combinatorics of the alternating sign matrix polytope $ASM_n$ using the elementary flow grid model and Brualdi–Dahl's doubly directed graphs. The authors prove that every face of $ASM_n$ is a 2-level polytope, characterize central symmetry of faces via vertex degrees in the doubly directed graph, and establish sharp bounds on the vertex and facet counts of faces in terms of dimension, including $v(F)\le 2^d$, $f(F)\le 4(d-1)$ and $vf\le d2^{d+1}$; they also show $v(F)\le 2^{d-1}+2$ when the graph is 2-connected. A key structural result is that no $ASM_n$ face has the combinatorial type of $B_3$, with a detailed analysis of ears and cycles guiding the bounds and decompositions. The paper develops cycle-basis descriptions of the affine hull, proves a product decomposition for certain faces, and catalogs low-dimensional combinatorial types of faces, linking ASM$n$ to broader 2-level polytope theory and enriching the combinatorial atlas of alternating sign matrices.

Abstract

The polytope $ASM_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $ASM_n$ 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 $ASM_n$ is symmetric if and only if its doubly directed graph has all vertices of even degree. We show that every face of $ASM_n$ is a 2-level polytope. We show that a $d$-dimensional face of $ASM_n$ has at most $2^d$ vertices and $4(d-1)$ facets, for $d\ge 2$. We show that a $d$-dimensional face of $ASM_n$ satisfies $vf\le d2^{d+1}$, 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 $ASM_n$ has the combinatorial type of the Birkhoff polytope $B_3$. We list the combinatorial types of faces of $ASM_n$ that have dimension 4 or less.

Figures (17)

• Figure 1: The simple flow grid of matrix $D_3$
• Figure 2: The complete flow grid $C_3$
• Figure 3: Vertex neighbourhoods in a simple flow grid
• Figure 4: Example of simple and elementary flow grids
• Figure 5: The union of three simple flow grids gives an elementary flow grid which would have a vertex adjacent to 3 double directed edges.

Theorems & Definitions (92)

• Definition 1.1
• Theorem 1.1
• Proposition 2.1
• Definition 2.1
• Definition 2.2
• Proposition 2.2
• proof
• Lemma 2.1
• proof
• Lemma 2.2