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Transitive factorizations in the symmetric group, and combinatorial aspects of singularity theory. (English) Zbl 0965.05003
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.

MSC:
05A05 Permutations, words, matrices
05E05 Symmetric functions and generalizations
05A15 Exact enumeration problems, generating functions
58C10 Holomorphic maps on manifolds
05A17 Combinatorial aspects of partitions of integers
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