Transitive powers of Young-Jucys-Murphy elements are central

TL;DR

This paper proves that transitive powers of Young-Jucys-Murphy elements are central in the symmetric group algebra, determines their coefficients, and explores implications for combinatorics, representation theory, and algebraic geometry.

Contribution

It introduces the concept of transitive powers of Young-Jucys-Murphy elements, proves their centrality, and analyzes the polynomiality of associated coefficients, linking to various mathematical areas.

Findings

Transitive powers are central in the group algebra of S_n.

Coefficients, called star factorization numbers, have polynomiality properties.

Results answer a question by Pak and extend symmetry results by Irving and Rattan.

Abstract

Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,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 centre 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 about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan; 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 associated with the star factorization numbers.