A note on the structure coefficients of the centraliser algebra

TL;DR

This paper generalizes the concept of partial permutations to construct a universal algebra that simplifies the proof of polynomiality and degree bounds of structure coefficients in m-centraliser algebras.

Contribution

It introduces a universal algebra framework that extends previous definitions and provides new proofs and bounds for structure coefficients in m-centraliser algebras.

Findings

New proof of polynomiality property

Upper bounds for polynomial degrees

Generalization of partial permutations

Abstract

In this note we generalize the definition of partial permutations of Ivanov and Kerov and we build a universal algebra which projects onto the m-centraliser algebra defined by Creedon. We use it to present a new proof for the polynomiality property of the structure coefficients of the m-centraliser algebra and to obtain upper bounds for the polynomial degrees.