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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.

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
Full Text: DOI
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