The hook fusion procedure and its generalisations

TL;DR

This paper introduces a new hook-based fusion procedure for the symmetric group, reducing auxiliary parameters, and extends it to Hecke algebras, aiding in constructing representations and verifying conjectures.

Contribution

It presents a novel hook fusion method that minimizes parameters and generalizes to ribbon decompositions and Hecke algebras for representation construction.

Findings

Efficient computation of diagonal matrix elements using hook fusion.

Supportive evidence for a previous conjecture.

Extension of the fusion procedure to Hecke algebras.

Abstract

The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically into ribbons. Finally, we adapt our construction to the quantum deformation of the symmetric group algebra known as the Hecke algebra.