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Семинар международной лаборатории кластерной геометрии "Характеристические классы и теория пересечений": Y. Burman

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Twisted Hurwitz numbers and Laplace--Beltrami operator

Classical Hurwitz numbers enumerate, among other things, graphs embedded into an oriented surface with boundary, with all the vertices on the boundary. This definition can be "twisted" allowing unoriented surfaces. The twisted Hurwitz numbers obtained collect into a generating function satisfying a PDE of Laplace--Beltrami family; another member of the family is the classical cut-and-join equation. This leads to an explicit formula for the twisted numbers involving zonal polynomials in the same way as classical Hurwitz numbers are expressed via Schur polynomials.

Algebraically, a classical Hurwitz number is, up to a factorial factor, the number of strings of transpositions in the permutation group S_n such that their product has a prescribed cyclic
type. Twisted Hurwitz numbers have a similar but more complicated description: the strings of transposition (in the group S_{2n}) are assumed to have a certain type of symmetry, and the product should belong to a particular conjugacy class of S_{2n} by its subgroup G isomorphic to the finite reflection group B_n.

This is a joint work with Raphael Fesler. We are going to discuss these definitions of the twisted Hurwitz numbers, as well as some other, e.g. the one involving ramified coverings (it was in fact known before our article appeared).