Nonstandard representations of type C affine Hecke algebra from K-operators

TL;DR

This paper develops new finite-dimensional representations of the type C affine Hecke algebra using solutions to the reflection equation derived from specific R-matrices related to quantum integrable models.

Contribution

It introduces a novel method to construct nonstandard representations of type C affine Hecke algebra via explicit solutions to the reflection equation based on Cremmer-Gervais and Jordanian R-matrices.

Findings

Constructed nonstandard representations for type C affine Hecke algebra.

Linked R-matrices solutions to the reflection equation with representation theory.

Extended the approach from type A to type C affine Hecke algebra.

Abstract

We construct nonstandard finite-dimensional representations of type C affine Hecke algebra from the viewpoint of quantum integrable models. There exists two classes of nonstandard solutions to the Yang-Baxter equation called the Cremmer-Gervais and Jordanian R-matrices. These R-matrices also satisfy the Hecke-relation, thus can be used to construct nonstandard finite-dimensional representations of type A affine Hecke algebra. We construct the corresponding nonstandard representations for type C affine Hecke algebra by explicitly constructing solutions to the reflection equation under the Hecke relation. We achieve this by taking the finite-dimensional representations and deBaxterizing the K-operators acting on the infinite-dimensional function space, taking advantage of the fact that the Cremmer-Gervais and Jordanian R-matrices can be obtained from the R-operator.