Cellularity of certain quantum endomorphism algebras

Research output: Working paperResearch

  • Henning Haahr Andersen, Denmark
  • G. I. Lehrer, University of Sydney, Australia
  • R. Zhang, University of Sydney, Australia
Let $\tA=\Z[q^{\pm \frac{1}{2}}][([d]!)\inv]$ and let
$\Delta_{\tA}(d)$ be an integral form of the Weyl module
of highest weight $d \in \N$ of the quantised enveloping algebra $\U_{\tA}$ of $\fsl_2$.
We exhibit for all positive integers $r$ an explicit cellular structure for $\End_{\U_{\tA}}(\Delta_{\tA}(d)^{\ot r})$. When $\zeta$ is a root of
unity of order bigger than $d$ we consider
the specialisation $\Delta_{\zeta}(d)^{\ot r}$ at $q\mapsto \zeta$
of $\Delta_{\tA}(d)^{\ot r}$. 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 $\U_{\zeta}(\fsl_2)$.
In the final section we independently determine the weight multiplicities of indecomposable tilting modules for $U_\zeta(\fsl_2)$ and the decomposition numbers of the endomorphism algebras. We indicate how our earlier results imply that either one of these sets of numbers determines the other.
Original languageDanish
Number of pages20
Publication statusPublished - 5 Mar 2013

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