CELLULARITY OF CERTAIN QUANTUM ENDOMORPHISM ALGEBRAS

For any ring (A) over tilde such that Z [q(+/- 1/2)] subset of (A) over tilde subset of Q(q(1/2)), let Delta((A) over tilde)(d) be an (A) over tilde -form of the Weyl module of highest weight d is an element of N of the quantised enveloping algebra U-(A) over tilde of sl(2). For suitable (A) over tilde, we exhibit for all positive integers r an explicit cellular structure for End U-(A) over tilde (Delta((A) over tilde) (d)(circle times r)). This algebra and its cellular structure are described in terms of certain Temperley-Lieb-like diagrams. We also prove general results that relate endomorphism algebras of specialisations to specialisations of the endomorphism algebras. When zeta is a root of unity of order bigger than d we consider the U-zeta-module structure of the specialisation Delta(zeta)(d)(circle times r) at q bar over right zeta of Delta((A) over tilde)(d)(circle times r). As an application of these results, we 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 U-zeta (sl(2)). As an example, in the final section we independently recover the weight multiplicities of indecomposable tilting modules for U-zeta (sl(2)) from the decomposition numbers of the endomorphism algebras, which are known through cellular theory.