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

Ian Goulden

Department of Combinatorics and Optimization
University of Waterloo

Abstract: Consider ordered factorizations of an arbitrary element of the symmetric group on n symbols, into factors which all belong to the same conjugacy class. An expression for the number of such factorizations is available via characters, but in certain cases a more compact answer can be obtained using a family of symmetric functions constructed by Macdonald. With the further restriction that the group generated by the factors acts transitively on the n symbols, we call these transitive factorizations. Hurwitz considered the number of transitive factorizations in the case that the factors are transpositions, because of its connection with counting all nonequivalent ramified coverings of a Riemann surface. In this case he gave a remarkably compact, explicit answer. In this talk, extensions of Hurwitz' result are presented, suggesting that transitive factorisations have an elegant structure, including a close, but as yet unknown, link with Macdonald's symmetric functions.