论文标题
匹配杰克猜想和法拉哈特·希格曼代数的完整性
Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra
论文作者
论文摘要
使用Jack多项式，Goulden和Jackson引入了一个一个参数变形的$τ_b$的一系列两部分映射，这概括了随机矩阵的$β$增强功能的分区函数。匹配杰克的猜想表明，在功率 - 基础上，功能$τ_b$的系数$ c^λ_{μ，ν} $是变形参数$ b $中的非负整数多项式。 dołęga和féray在2016年证明了匹配杰克猜想中的“多项式”部分，即，系数$ c^λ_{μ{μ，ν} $在$ \ mathbb {q} [q} [b] $中。在本文中，我们证明了“完整性”部分，即系数$ c^λ_{μ，ν} $在$ \ mathbb {z} [b] $中。 证明是基于作者的最新作品，该作者从模拟结果中推论了$ b $ conconture的边际总和的匹配杰克猜想，该结果由Chapuy anddołęga于2020年建立。证明的关键步骤涉及与分级的Farahat-Higman代数建立新的联系。
Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation $τ_b$ of the generating series of bipartite maps, which generalizes the partition function of $β$-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients $c^λ_{μ,ν}$ of the function $τ_b$ in the power-sum basis are non-negative integer polynomials in the deformation parameter $b$. Dołęga and Féray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely that coefficients $c^λ_{μ,ν}$ are in $\mathbb{Q}[b]$. In this paper, we prove the "integrality" part, i.e that the coefficients $c^λ_{μ,ν}$ are in $\mathbb{Z}[b]$. The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums from an analog result for the $b$-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra.
