发表机构
Drexel University; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications; Stockholm University（德雷塞尔大学; 北京雁栖湖应用数学研究院; 斯德哥尔摩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图的匹配代数的 Dunkl 子代数，证明其与匹配 Specht 生成元对偶，并刻画饱和性与连通性的关系，给出完全图、森林和单圈图的 Hilbert 级数。
AI 中文摘要
对于有限简单图 $G = (V, E)$，我们考虑多项式环在变量 $u_e$（其中 $e \in E$）上由所有非不相交边 $e$ 和 $f$ 的乘积 $u_e u_f$（包括所有平方 $u_e^2$）生成的理想所得到的商代数 $\mathcal M(G)$。该商称为 $G$ 的匹配代数，因为它有一个由 $G$ 的匹配索引的基。在此商中，我们定义一个由所有 $v \in V$ 的带符号关联和 $\theta_v=\sum_{e=(v,w)}u_e-\sum_{e=(w,v)}u_e$（其中 $G$ 的所有边任意定向）生成的子代数 $\mathcal D(G)$；我们称之为 Dunkl 匹配代数。我们证明，作为分次向量空间，$\mathcal D(G)$ 与所有多项式 $p_M = \prod_{(i,j) \in M} (x_i - x_j)$ 的跨度是对偶的，其中 $M$ 遍历 $G$ 的所有匹配。对于完全图 $K_n$，后者的跨度是两个行 Specht 模（每个次数一个）的直和；因此其 Hilbert 级数是 Catalan 三角形的级数，并且在特征零的情况下，Dunkl 匹配代数可以由线性和二次关系给出。对于任意图，我们提出了饱和问题，即决定在给定次数 $k$ 中选定的匹配 Specht 生成元 $p_M$ 是否张成完整的两个行 Specht 模 $S^{(n-k,k)}$。我们证明次数 $k$ 的饱和蕴含 $k$-连通性，次数 $2$ 的饱和等价于 $2$-连通性，并且 $k$-连通图在次数 $k$ 是饱和的。我们还证明当 $G$ 的补图是匹配时，在每个可能的次数都是饱和的。我们证明 Dunkl 匹配代数恰好等于完整匹配代数当且仅当 $G$ 是森林，并给出单圈图的显式 Hilbert 级数公式。
英文摘要
For a finite simple graph $G = (V, E)$, we consider the quotient algebra $\mathcal M(G)$ of the polynomial ring in the variables $u_e$ (for $e \in E$) by the ideal generated by all products $u_e u_f$ for non-disjoint edges $e$ and $f$ (including all squares $u_e^2$). This quotient is called the matching algebra of $G$, since it has a basis indexed by the matchings of $G$. In this quotient, we define a subalgebra $\mathcal D(G)$ generated by the signed incidence sums $θ_v=\sum_{e=(v,w)}u_e-\sum_{e=(w,v)}u_e$ for all $v \in V$ (where all edges of $G$ are oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space, $\mathcal D(G)$ is dual to the span of all polynomials $p_M = \prod_{(i,j) \in M} (x_i - x_j)$, where $M$ ranges over all matchings of $G$. For the complete graph $K_n$, the latter span is a direct sum of two-row Specht modules (one in each degree); thus its Hilbert series is that of the Catalan triangle, and, in characteristic zero, the Dunkl matching algebra can be presented by linear and quadratic relations. For arbitrary graphs, we formulate the saturation problem of deciding when the selected matching Specht generators $p_M$ in a given degree $k$ span the full two-row Specht module $S^{(n-k,k)}$. We show that saturation in degree $k$ forces $k$-connectivity, that saturation in degree $2$ is equivalent to $2$-connectivity, and that $k$-linked graphs are saturated in degree $k$. We also prove saturation in every possible degree whenever the complement of $G$ is a matching. We show that the Dunkl matching algebra equals the full matching algebra exactly for forests, and give an explicit Hilbert series formula for unicyclic graphs.
Comments32 pages. Produced using GPT-5.6 based on third author's notes; significantly edited and fully proofread. v2 adds Motivation subsection on pp. 3--4, a new Remark 3.6 on characteristic dependence, and a new Section 6 with log-concavity conjectures. Comments are welcome!
