A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman–Wenzl, and Type A Iwahori–Hecke Algebras

Abstract

We show how the ribbon Hopf algebra structure on the Drinfel'd Jimbo quantum groups of Types A, B, C, and D can be used to derive formulas giving explicit realizations of the irreducible representations of the Iwahori Hecke algebras of type A and the Birman Wenzl algebras. We use this derivation to give explicit realizations of the irreducible representations of the Brauer algebras as well. The derivation is accomplished by way of a combination of techniques from operator algebras, quantum groups, and the theory of 3-manifold invariants. Although our applications are in the cases of the quantum groups of Types A, B, C, and D, most of the aspects of our approach apply in the general setting of ribbon Hopf algebras.

On Brauer algebra simple modules over the complex field

“…It turns out that the combinatorics used in [15] and [11] can be explained in a uniform and natural way in this geometrical context. In particular, we obtain a striking characterisation of the roots of the King polynomials.…”

“…Recall that the Young graph Y has, as vertex set, the set Λ of all partitions, and two partitions λ and µ are connected by an edge if λ µ or λ µ. Leduc and Ram constructed in [11] bases for the standard modules for the Brauer algebra for generic values of δ in terms of walks on Y. Their construction relies on complex combinatorial objects such as the King polynomials (first introduced in [7]).…”

“…Motivated by this, the second objective of this paper is to recast the contruction of [11] in the geometrical setting. This provides a unification of the classical and modern approaches, but is also done with a view to treating arbitrary simple modules (and other characteristics) in further work.…”

“…Note that in all cases the vertices i and j in λ must be in the same chamber otherwise x µ wouldn't have the required cap diagram. We have that cases D(i)(iii)-(v) correspond to all vertices j satisfying (11), and Cases D(ii) and (vi) correspond to all vertices j satisfying (12). Finally, using the fact that all simple modules are self-dual, the restriction must have the required module structure.…”

“…In this section we recall the construction given in [11] of walk bases for simple modules for the generic Brauer algebra B n (u), where u is an indeterminate and the algebra is defined over C(u). Their construction uses two combinatorial objects associated with partitions which we now recall.…”