Transitive powers of Young–Jucys–Murphy elements are central

Although powers of the Young–Jucys–Murphy elements , i=1,…,n, in the symmetric group Sn acting on {1,…,n} do not lie in the center of the group algebra of Sn, we show that transitive powers, namely the sum of the contributions from elements that act transitively on [n], are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the center of Sn, and show that they have a polynomiality property. These centrality and polynomiality properties have seemingly unrelated consequences. First, they answer a question raised by Pak [I. Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ary trees, Discrete Math. 204 (1999) 329–335] about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan [J. Irving, A. Rattan, Minimal factorizations of permutations into star transpositions, Discrete Math., in press, math.CO/0610640]; and thirdly, we relate the polynomiality to an existing polynomiality result for a class of double Hurwitz numbers associated with branched covers of the sphere, which therefore suggests that there may be an ELSV-type formula (see [T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001) 297–327]) associated with the star factorization numbers.