I’m reading the following definitions and theorem on alternating sign matrices (ASMs).
Definition 1. Alternating sign matrices (ASMs) are square matrices with the following properties:
- entries $\in\{0,1,-1\}$
- the entries in each row and column sum to 1
- nonzero entries in each row and column alternate in sign
Definition 2. The $n$th alternating sign matrix polytope, which we will denote as $ASM_{n}$, is the convex hull in $\mathbb{R}^{n^{2}}$ of the $n \times n$ alternating sign matrices.
By previous fact gathering, I know that $ASM_n$ has dimension $(n-1)^2$, and satisfies the following defining relations:
\begin{align} \sum_{i^\prime = 1}^{i} x_{i^\prime j} \geq 0 \quad & \forall i,j \in [n] \\ \sum_{i^\prime = i}^{n} x_{i^\prime j} \geq 0 \quad & \forall i,j \in [n] \\ \sum_{j^\prime = 1}^{j} x_{i j^\prime} \geq 0 \quad & \forall i,j \in [n] \\ \sum_{j^\prime = j}^{n} x_{i j^\prime} \geq 0 \quad & \forall i,j \in [n] \\ \sum_{i=1}^{n} x_{i j}=1 \quad & \forall j \in [n] \\ \sum_{j=1}^{n} x_{i j}=1 \quad & \forall i \in [n] \end{align}
(The first four conditions say that partial sums (i.e., sums of first $i$ / last $n-i$ entries of a column, or sums of first $j$ / last $n-j$ entries of a row) are non-negative, while the last two conditions say that column and row entries sum to $1$.)
We find the facets by setting the above inequalities to be equalities, and see which of them yields a facet. Some further analysis leads to this paragraph.
Thus $A S M_{n}$ has at most $4\left[(n-2)^{2}+1\right]$ facets, given explicitly by the $4(n-2)^{2}+4$ sets of all $X \in A S M_{n}$ which satisfy one of the following: $$ \begin{array}{c} \sum_{i^{\prime}=1}^{i-1} x_{i^{\prime} j}=0, \sum_{j^{\prime}=1}^{j-1} x_{i j^{\prime}}=0, \sum_{i^{\prime}=i+1}^{n} x_{i^{\prime} j}=0, \sum_{j^{\prime}=j+1}^{n} x_{i j^{\prime}}=0, i, j \in\{2, \ldots, n-1\}, \\ x_{11}=0, x_{1 n}=0, x_{n 1}=0, \text { or } x_{n n}=0 . \end{array} $$ They are facets (not just faces) since each equality determines exactly one more entry of the matrix, decreasing the dimension by one.
I have trouble understanding the last statement. Would be great if someone could elaborate.