Characters of algebras containing a Jones basic construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras
type: Item
instance of: scholarly article
title: Characters of algebras containing a Jones basic construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras
zbMATH Open document ID: 0856.16038
author: Arun Ram
author: Tom Halverson
published in: Advances in Mathematics
publication date: 1996-11-04T00:00:00Z
review text: If \(A\) and \(B\) are split semisimple algebras such that \(A\) is a subalgebra of \(B\), the centralizer of the action of \(A\) on \(B\) by left multiplication, \(C=\text{End}_A(B)\), is called the Jones basic construction for the inclusion \(A\subseteq B\). The paper is devoted to computation of characters of algebras containing the basic construction, namely of Temperley-Lieb, Brauer, Birman-Wenzl and Okada algebras. A general theorem is given which says that the characters of \(C=\text{End}_A(B)\) are completely determined by the characters of the subalgebra \(A\). The characters of irreducible \(C\) are determined in general form and used to compute the characters of the Temperley-Lieb algebras and the Okada algebras. To determine the characters of the Birman-Wenzl (B.-W.) algebras, the theory of quantum groups and the quantum Yang-Baxter equation are exploited. In addition to the characters which are obtained due to the general theorem the others are computed by using the duality between B.-W. algebras and Drinfeld-Jimbo (1986) quantum groups of certain types. Certain characters of the quantum groups are obtained which are polynomials invariant under the Weyl group and can be viewed as generalizations of the power symmetric functions and of certain Hall-Littlewood polynomials. It is shown that the characters of the B.-W. algebras may be derived as the coefficients in the expansions of these polynomials in terms of the Weyl characters. As a result, combinatorial rules for computing the irreducible characters of B.-W. algebras are given which may be also used as special cases for computation of the irreducible characters for Iwahori-Hecke (1964) and the Brauer (Ram, 1991) algebras.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 863620
zbMATH Keywords: Birman-Wenzl algebras
zbMATH Keywords: Iwahori-Hecke algebras
zbMATH Keywords: Brauer algebras
zbMATH Keywords: semisimple algebras
zbMATH Keywords: centralizers
zbMATH Keywords: actions
zbMATH Keywords: Jones basic construction
zbMATH Keywords: characters of algebras
zbMATH Keywords: Okada algebras
zbMATH Keywords: Temperley-Lieb algebras
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: Weyl groups
zbMATH Keywords: symmetric functions
zbMATH Keywords: Hall-Littlewood polynomials
zbMATH Keywords: irreducible characters
reviewed by: A. A. Bogush
MaRDI profile type: Publication
full work available at URL: aima.1995.1068
OpenAlex ID: W1990992940
DOI: 10.1006/AIMA.1995.1068
description: scientific article; zbMATH DE number 863620