Transitive powers of Young–Jucys–Murphy elements are central

Abstract

Although powers of the Young–Jucys–Murphy elements X i = ( 1 i ) + ( 2 i ) + ⋯ + ( i − 1 i ) , i = 1 , … , n , in the symmetric group S n acting on { 1 , … , n } do not lie in the center of the group algebra of S n , 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 S n , 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.