Characters of Algebras Containing a Jones Basic Construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl Algebras

Abstract

We begin by determining, in a general form, the characters of irreducible representations of a Jones basic construction and use this result to compute the characters of the Temperley-Lieb algebras and the Okada algebras. In the case of the Birman-Wenzl algebra some of the characters are determined by the general theorem and the others are computed by using the duality between the Birman-Wenzl algebras and the Drinfeld-Jimbo quantum groups of types B and C. The computations involve certain characters of the quantum group; these are polynomials invariant under the Weyl group of type B. We are able to decompose these Weyl group symmetric functions and obtain a combinatorial rule for computing the irreducible characters of the Birman-Wenzl algebras. The combinatorial rules for computing the irreducible characters of the Iwahori-Hecke algebras and the combinatorial rule for computing the irreducible characters of the Brauer algebras are both special cases of our rule for the Birman-Wenzl algebras.