Some remarks on a result of Jensen and tilting modules for $\SL_3(k)$ and $q$-$\GL_3(k)$

Details Some remarks on a result of Jensen and tilting modules for $\SL_3(k)$ and $q$-$\GL_3(k)$ Some remarks on a result of Jensen and tilting modules for $\SL_3(k)$ and $q$-$\GL_3(k)$

2008-09-12

arxiv.org/abs/0809.2249v2

Mathematics

Group Theory

18 pages, 8 figures

Scientific paper

This paper reviews a result of Jensen on characters of some tilting modules for $\SL_3(k)$, where $k$ has characteristic at least five and fills in some gaps in the proof of this result. We then apply the result to finding some decomposition numbers for three part partitions for the symmetric group and the Hecke algebra. We review what is known for characteristic two and three. The quantum case is also considered: analogous results hold for the mixed quantum group where $q$ is an $l$th root of unity with $l$ at least three and thus also hold for the associated Hecke algebra.

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