Towards studying the structure of triple Hurwitz numbers

Abstract

Going beyond the studies of single and double Hurwitz numbers, we report some progress towards studying Hurwitz numbers which correspond to ramified coverings of the Riemann sphere involving three nonsimple branch points. We first prove a recursion which implies a fundamental identity of Frobenius enumerating factorizations of a permutation in group algebra theory. We next apply the recursion to study Hurwitz numbers involving three nonsimple branch points (besides simple ones), two of them having complete ramification profiles while the remaining one having a prescribed number of preimages. The recursion allows us to obtain recurrences as well as explicit formulas for these numbers. The case where one of the nonsimple branch points with complete profile has a unique preimage (one-part quasi-triple Hurwitz numbers) is particularly studied in detail. We prove a dimension-reduction formula from which any one-part quasi-triple Hurwitz number can be reduced to quasi-triple Hurwitz numbers where two branch points respectively have a unique preimage. We also obtain the polynomiality of one-part quasi-triple Hurwitz numbers analogous to that implied by the remarkable ELSV formula, suggesting a potential connection to Hodge integrals or intersection theory. Coefficients of the polynomials are completely and explicitly determined which may facilitate searching these integral counterparts (i.e., ELSV-type formulas) in the future.Comment: An extended abstract, comments are very much welcome. The full version with more results and details will be uploaded late