6533b7ddfe1ef96bd127364a

RESEARCH PRODUCT

Transitive factorizations in the hyperoctahedral group

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The classical Hurwitz enumeration problem has a presentation in terms of transitive factor- izationsin the symmetric group. This presentationsuggestsageneralizationfromtypeAto otherfinite reflection groups and, in particular, to type B.W e study this generalization both from ac ombinatorial and a geometric point of view, with the prospect of providing am eans of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. W ec onjecture an analogous setting for the type B case that is studied here. 1I ntroduction Transitive factorizations of permutations into transpositions occur in Hurwitz's ap- proach to determining the Hurwitz number hg(θ), the number of genus g ramified covers of the sphere with elementary branching at a prescribed number of points and arbitrary ramification over infinity specified by the partition θ .T his problem, which is called Hurwitz's enumeration problem, and its generalizations have attracted con- siderable attention in recent years and have been shown to have deep connections through geometry to the moduli space of maps. The presentation of Hurwitz's prob- lem in terms of factorizations of permutations makes it susceptible to approaches from algebraic combinatorics, and these approaches have assisted our understanding ofthe problem. ThetypeAsetting oftheproblemstronglysuggestsanotherdirection of generalization, namely to other finite reflection groups. In this paper we study the Hurwitz problem for type B ,t he hyperoctahedral group, with the purpose of under- standing both the combinatorics and the geometry of this generalization. We derive the main result by combinatorial means and then provide a geometrical explanation of the resultso that the connection between the two approachesmay be better under- stood. We surmise that the S2-action that is present in the type B Hurwitz problem might yield new moduli spaces of maps with a specific S2-symmetry. The organization of the paper is as follows (the few undefined terms appearing in this paragraph are defined later). In Section 2, we give the axiomatization of admis- sible and near-admissible factorizations for the hyperoctahedral Hurwitz numbers in terms of permutation factorizations, and give combinatorial properties of the hyper- octahedral group. InSection 3, wesolvethehyperoctahedralanalogueoftheHurwitz problembyenumeratingan auxiliary setof factorizations called admissible factoriza- tions. This is our main combinatorial result, stated in Theorem 3.4, which expresses