Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
Guillaume Chapuy, Maciej Dołęga
备注：
56 pages, 6 figures; v2: definition of generalized branched covers fixed; combinatorial decomposition and corresponding equations now presented for connected objects and duality introduced; proof of piecewise polynomiality changed accordingly; v3: minor corrections
我们引入了一个单参数变形的2-Toda τ-函数的（加权）Hurwitz数，通过变形Schur函数到杰克对称函数。我们证明了它的系数是关于变形参数$b$的多项式，系数为非负整数。这些系数计数广义分支覆盖的领域的任意表面，定向或不定向，与适当的$b$加权，“措施”在某种意义上说，他们的不可定向性。值得注意的特殊情况包括不可定向的dessins d 'enfants，我们证明了迄今为止最普遍的结果，对匹配杰克猜想和“$b$ -猜想”的Goulden和Jackson从1996年，扩展的$β$ -系综矩阵模型，变形的HCIZ积分，和$b$ -Hurwitz数，我们在这里介绍，这是$b$ -变形的经典（单或双）Hurwitz数得到的$b=0$。在我们的证明中发挥了关键作用的组合模型的不可定向星座配备了一个合适的$b$加权，其分拆函数满足一个无限集的偏微分方程。这些偏微分方程有两个定义，一个由Lax方程给出，另一个遵循显式组合分解。
英文摘要：
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.
Weighted Hurwitz numbers and hypergeometric $τ$-functions: an overview
This is an overview of recent results on the use of 2D Toda $τ$ -functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$ . Specializing the fermionic formula to the case of diagonal group elements gives $τ$ -functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of…
