A new fusion procedure for the Brauer algebra and evaluation homomorphisms

A. P. Isaev, A. I. Molev and O. V. Ogievetsky

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.