Jucys-Murphy elements and a presentation for partition algebras

TL;DR

This paper introduces a new presentation and inductive formula for partition algebras and their Jucys-Murphy elements, facilitating the construction of seminormal representations.

Contribution

It provides a novel presentation and recursive formula for partition algebra Jucys-Murphy elements, linking them via Schur-Weyl duality to existing definitions.

Findings

New presentation for partition algebras

Equivalent recursive formula for Jucys-Murphy elements

Foundation for constructing seminormal representations

Abstract

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.