Representation Theory and Mathematical Physics Seminar: Generating rationally weighted Hurwitz numbers with KP \(\tau \)-functions

Seminar | April 26 | 3-4 p.m. | 740 Evans Hall

 John Harnad, CRM Montreal and Concordia University

 Department of Mathematics

Hurwitz numbers enumerate branched coverings of the Riemann sphere with specified branching profiles. \(\tau \)-functions of hypergeometric type for the KP and \(2D\)-Toda integrable hierarchies serve as combinatorial generating functions for weighted sums over Hurwitz numbers, with weights chosen as symmetric functions of a set of auxiliary parameters determined by a weight generating function. This talk will explain how multicurrent correlators may be used to explicitly generate weighted Hurwitz numbers as weighted polyonomials in the Taylor coefficients of the weight generating function, without any knowledge required either of symmetric group characters or the Kostka matrices relating different bases of the ring of symmetric functions. The case of rational weight generating functions will be the main illustrative example.