A new fusion procedure for the Brauer algebra and evaluation homomorphisms

TL;DR

This paper introduces a novel fusion procedure for the Brauer algebra using R-matrix evaluations, linking it to symmetric groups and classical Lie algebra representations, and constructs an evaluation homomorphism connecting reflection algebras to universal enveloping algebras.

Contribution

It presents a new fusion method for the Brauer algebra and symmetric group involving R-matrices, and establishes an evaluation homomorphism between reflection algebra and universal enveloping algebra.

Findings

Primitive idempotents obtained via rational function evaluation.

Fusion procedure connects reflection algebra representations with evaluation modules.

Provides a new perspective on the relationship between Brauer algebra and Lie algebra representations.

Abstract

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provides an equivalence between natural tensor representations of B(g_N) with the corresponding evaluation modules.