Cellularity of certain quantum endomorphism algebras

TL;DR

This paper constructs explicit cellular structures for quantum endomorphism algebras associated with tensor powers of Weyl modules, linking their simple modules to tilting module weight multiplicities, especially at roots of unity.

Contribution

It provides a new cellularity framework for quantum endomorphism algebras and relates their simple modules to tilting module weight multiplicities at roots of unity.

Findings

Cellularity structures for all positive integers r.

Conditions for commutativity of specialization and endomorphism algebra formation.

Equivalence between simple module dimensions and tilting module weight multiplicities.

Abstract

We exhibit for all positive integers r, an explicit cellular structure for the endomorphism algebra of the r'th tensor power of an integral form of the Weyl module with highest weight d of the quantised enveloping algebra of sl2. When q is specialised to a root of unity of order bigger than d, we consider the corresponding specialisation of the tensor power. We prove one general result which gives sufficient conditions for the commutativity of specialisation with the taking of endomorphism algebras, and another which relates the multiplicities of indecomposable summands to the dimensions of simple modules for an endomorphism algebra. Our cellularity result then allows us to prove that knowledge of the dimensions of the simple modules of the specialised cellular algebra above is equivalent to knowledge of the weight multiplicities of the tilting modules for the specialised quantum group. In the final section we independently determine the weight multiplicities of indecomposable tilting modules for quantum sl2, and the decomposition numbers of the endomorphism algebras. We indicate how either one of these sets of numbers determines the other.