Research output: Working paper › Research

- 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.

$\Delta_{\tA}(d)$ 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 language | Danish |
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Publisher | arXiv.org |

Edition | 1303.0984 |

Number of pages | 20 |

Publication status | Published - 5 Mar 2013 |

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