Transitive factorisations in the symmetric group, and combinatorial aspects of singularity theory

Abstract

We consider the determination of the number $c_k(α)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $α$, into k-cycles such that the factorisations have minimal length and such that the group generated by the factors acts transitively on the n symbols. The case k=2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for $c_k(α)$, and prove that it holds for factorisations of permutations with one, two and three cycles ($α$ is a partition with at most three parts). The generating series is naturally expressed in terms of the symmetric functions dual to those introduced by Macdonald for the ``top'' connection coefficients in the class algebra of the symmetric group. Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.