Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 097, 19 pages      arXiv:1504.07512
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Multispecies Weighted Hurwitz Numbers

J. Harnad ab
a) Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada
b) Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montréal (QC) H4B 1R6, Canada

Received March 31, 2015, in final form November 16, 2015; Published online December 02, 2015; Misprints corrected December 10, 2015

The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.

Key words: weighted Hurwitz number; $\tau$-function; multispecies.

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