# 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.

## Key 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.