Invariant tensors and cellular categories

TL;DR

This paper proves that the integral form of the centraliser algebra associated with tensor products of highest weight representations of a quantised enveloping algebra is cellular, using Lusztig's results.

Contribution

It establishes the cellularity of the integral form of the centraliser algebra in the context of quantised enveloping algebras of finite type.

Findings

The dual canonical basis provides an integral form of the centraliser algebra.

The integral form is shown to be cellular.

Uses Lusztig's results to prove cellularity.

Abstract

Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then the centraliser algebra of W has a basis called the dual canonical basis which gives an integral form. We show that this integral form is cellular by using results due to Lusztig.