论文标题
不可取向的分支覆盖物，$ b $ -Hurwitz号码和多参数插孔的积极性
Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
论文作者
论文摘要
我们引入了（加权）Hurwitz数字的2-TADA TAU功能的一参数变形，该功能通过将Schur函数变形为插孔对称函数而获得。我们表明，其系数是带有非负整数系数的变形参数$ b $中的多项式。这些系数通过任意表面（无论是否定向）对球体的总体分支覆盖率进行计算，并以适当的$ b $加权从某种意义上说“测量”。 值得注意的特殊情况包括不可定向的Dessins d'Enfant，我们证明了迄今为止最终取向匹配杰克的猜想和1996年Goulden和Jackson的“ $ b $ conconture”和$β$ - 符号矩阵模型的扩展，HCIZ Integrials of HCIZ Integrial和$ b $ - $ - $ - $ - $ -HURWWITZ数字，我们介绍了我们的构成，我们的成绩是我们的构成，我们的成绩是我们的构建。 （单或双）为$ B = 0 $获得的Hurwitz号码。 在我们的证明中的关键作用是由配备合适的$ b $加权的非定向星座组合模型扮演的，其分区功能满足了无限的PDES。这些PDE具有两个定义，一个由LAX方程给出，另一个是遵循明确组合分解的定义。
We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the "$b$-conjecture" of Goulden and Jackson from 1996, expansions of the $β$-ensemble matrix model, deformations of the HCIZ integral, and $b$-Hurwitz numbers that we introduce here and that are $b$-deformations of classical (single or double) Hurwitz numbers obtained for $b=0$. A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable $b$-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.
