Polynomiality of monotone Hurwitz numbers in higher genera
| Autor: | Jonathan Novak, Ian P. Goulden, Mathieu Guay-Paquet |
|---|---|
| Rok vydání: | 2013 |
| Předmět: |
05A15
14E20 (Primary) 15B52 (Secondary) Discrete mathematics Hurwitz quaternion 010308 nuclear & particles physics Mathematics::Number Theory General Mathematics 010102 general mathematics Klein quartic Riemann sphere 01 natural sciences Routh–Hurwitz stability criterion Combinatorics symbols.namesake Mathematics::Algebraic Geometry Monotone polygon 0103 physical sciences Hurwitz's automorphisms theorem FOS: Mathematics symbols Mathematics - Combinatorics Hurwitz matrix Combinatorics (math.CO) Hurwitz polynomial 0101 mathematics Mathematics |
| Zdroj: | Advances in Mathematics. 238:1-23 |
| ISSN: | 0001-8708 |
| DOI: | 10.1016/j.aim.2013.01.012 |
| Popis: | Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition. Comment: 23 pages |
| Databáze: | OpenAIRE |
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