Counting 3-way contingency tables via quiver semi-invariants
University of Missouri-Columbia（密苏里大学哥伦比亚分校）；International Center for Mathematical Sciences, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所国际数学科学中心）
令 T_{a,b} 表示大小为 m×n×p 的三向 contingency 表，其中两个平面边和固定由 a=(a₁,…,aₘ)∈ℕ^m 和 b=(b₁,…,bₙ)∈ℕ^n 给定。当 p=1 时，这表示 m×n 非负整数矩阵的行和列和固定由 a 和 b 给定的数目。本文通过有向图不变量理论研究 T_{a,b}，令 Q^{p}_{m,n} 为 p-完全 bipartite 有向图，具有 m 个源顶点，n 个汇顶点，以及每个源到每个汇 p 条箭头。令 1 表示 Q^{p}_{m,n} 的维度向量，取值为 1 在每个顶点。令 θ_{a,b} 表示分配 a_i 到第 i 个源顶点，-b_j 到第 j 个汇顶点的整数权重。本文首先将 T_{a,b} 视为 (Q^{p}_{m,n}, 1, θ_{a,b}) 关联的半不变量空间的维数。利用这一联系和有向图不变量理论的方法，证明 T_{a,b} 是抛物线 Kostka 系数。在 p=1 情况下，这恢复了行和列和固定由 a 和 b 给定的 m×n contingency 表的数目，经典 2-向情况也可通过 Robinson-Schensted-Knuth 对应得到。
英文摘要：
Let $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ be the number of $3$-way contingency tables of size $m \times n \times p$ with two of its three plane-sum margins fixed by $\mathbf{a}=(a_1, \ldots, a_m) \in \mathbb{N}^m$ and $\mathbf{b}=(b_1, \ldots, b_n) \in \mathbb{N}^n$. When $p=1$, this is the number of $m \times n$ non-negative integer matrices whose row and column sums are fixed by $\mathbf{a}$ and $\mathbf{b}$.
In this paper, we study the numbers $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ through the lens of quiver invariant theory. Let $\mathcal{Q}^{p}_{m,n}$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $\mathbf{1}$ denote the dimension vector of $\mathcal{Q}^{p}_{m,n}$ that takes value $1$ at every vertex of $\mathcal{Q}^{p}_{m,n}$, and let $θ_{\mathbf{a}, \mathbf{b}}$ denote the integral weight that assigns $a_i$ to the $i^{th}$ source vertex and $-b_j$ to the $j^{th}$ sink vertex of $\mathcal{Q}^{p}_{m,n}$.
We begin by realizing $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $(\mathcal{Q}^{p}_{m,n}, \mathbf{1}, θ_{\mathbf{a}, \mathbf{b}})$. Using this connection and methods from quiver invariant theory, we show that $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ is a parabolic Kostka coefficient. In the case $p=1$, this recovers the formula for the number of the $m \times n$ contingency tables with row and column sums fixed by $\mathbf{a}$ and $\mathbf{b}$, which in the classical $2$-way setting can also be obtained via the Robinson-Schensted-Knuth correspondence.
