an:01545623
Zbl 0965.05003
Goulden, Ian P.; Jackson, David M.
Transitive factorizations in the symmetric group, and combinatorial aspects of singularity theory
EN
Eur. J. Comb. 21, No. 8, 1001-1016 (2000).
00070777
2000
j
05A05 05E05 05A15 58C10 05A17
transitive factorizations of permutations; combinatorics of symmetric groups; combinatorial aspects of singularity theory
The paper studies the number \(c_k(\alpha)\) of ordered factorizations of an arbitrary permutation on \(n\) symbols, with cycle distribution \(\alpha\), into \(k\)-cycles, such that the factorizations have minimal length and such that the group generated by the factors acts transitively on the \(n\) symbols. Such factorizations are encountered in a number of contexts as topological classification of polynomials of a given degree and a given number of critical values, the moduli space of covers of the Riemann sphere and properties of the Hurwitz monodromy group, applications to mathematical physics. For example, 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 (and the monodromy group is the full symmetric group). When \(k=3\), the monodromy group is the alternating group and this case is also of considerable interest. The authors conjecture an explicit form for the generating series of \(c_k(\alpha)\) for any \(k\). They prove their conjecture for factorizations with one, two and three cycles, i.e. when \(\alpha\) is a partition with at most three parts. A striking common element between the results of this paper on transitive minimal ordered factorizations and Macdonald's ``top'' symmetric functions is the functional equation \(w=x\cdot\text{ exp}(w^{k-1})\) that arises in both settings when \(k\)-cycles are factors, for apparently different reasons.
Vesselin Drensky (Sofia)