Partial Jucys-Murphy elements and star factorizations

TL;DR

This paper explores factorizations of permutations into star transpositions, revealing hidden symmetries, providing new proofs, and connecting to algebraic structures using partial Jucys-Murphy elements.

Contribution

It introduces natural analogs of Jucys-Murphy elements in partial permutation algebras and explains symmetries in permutation factorizations with new proofs and expansions.

Findings

Identified hidden symmetry in star transposition factorizations

Provided a new proof of Goulden and Jackson's explicit formula

Derived normalized class expansion of central elements in symmetric group algebra

Abstract

In this paper, we look at the number of factorizations of a given permutation into star transpositions. In particular, we give a natural explanation of a hidden symmetry, answering a question of I.P. Goulden and D.M. Jackson. We also have a new proof of their explicit formula. Another result is the normalized class expansion of some central elements of the symmetric group algebra introduced by P. Biane. To obtain this results, we use natural analogs of Jucys-Murphy elements in the algebra of partial permutations of V. Ivanov and S. Kerov. We investigate their properties and use a formula of A. Lascoux and J.Y. Thibon to give the expansion of their power sums on the natural basis of the invariant subalgebra.