Characters of the centralizer algebras of mixed tensor representations of GL(r, ℂ) and the quantum group 𝒰_q(gl(r, ℂ))

Tom Halverson

Vol. 174 (1996), No. 2, 359–410

DOI: 10.2140/pjm.1996.174.359

Abstract

We consider the tensor product representation of m copies of the natural representation with n copies of its dual representation for both the general linear group GL(r, ℂ) and the quantum group U_q(gℓ(r, ℂ)). These tensor spaces determine rational representations of GL_r and 𝒰_q(gℓ(r, ℂ)). The centralizer algebras of these representations are, respectively, the complex algebra ℬ_m,n^r, which is a subalgebra of the Brauer algebra B_m+n^r, and the algebra H_m,n^r(q) over the field of complex rational functions with indeterminate q, which is a generalization of the Iwahori-Hecke algebra. Upon setting q = 1, the algebra H_m,n^r(q) specializes to ℬ_m,n^r. The algebra ℬ_m,n^r contains as a subalgebra the group algebra ℂ[S_m × S_n] of the product of two symmetric groups, and the algebra H_m,n^r(q) contains as a subalgebra the tensor product H_m(q) ⊗ H_n(q) of two Iwahori-Hecke algebras. In each centralizer, we find a distinguished basis and define an analog of conjugacy class. We then exploit Schur’s double centralizer theory to derive a “Frobenius formula” which we use to compute their irreducible characters in terms of symmetric group characters and Iwahori-Hecke algebra characters. In the process, we obtain branching rules that give the decomposition of ℬ_m,n^r-modules into irreducible ℂ[S_m × S_n]-modules and H_m,n^r(q)-modules into irreducible H_m(q) ⊗ H_n(q)-modules.

Mathematical Subject Classification 2000

Primary: 17B37

Milestones

Received: 1 July 1993

Published: 1 June 1996