Murphy elements from the double-row transfer matrix

TL;DR

This paper investigates the double-row transfer matrix in integrable lattice models, expressing it through Hecke algebra generators and extracting Murphy elements, which commute under certain conditions, revealing algebraic structures in these models.

Contribution

It introduces a method to express the open transfer matrix using Hecke algebra generators and identifies Murphy elements within this framework, linking algebraic structures to integrable models.

Findings

Murphy elements are extracted from the transfer matrix expansion.

Certain combinations of Murphy elements commute with the Hecke algebra.

The approach connects algebraic elements to boundary conditions in lattice models.

Abstract

We consider the double-row (open) transfer matrix constructed from generic tensor-type representations of various types of Hecke algebras. For different choices of boundary conditions for the relevant integrable lattice model we express the double-row transfer matrix solely in terms of generators of the corresponding Hecke algebra (tensor-type realizations). We then expand the open transfer matrix and extract the associated Murphy elements from the first/last terms of the expansion. Suitable combinations of the Murphy elements as has been shown commute with the corresponding Hecke algebra.