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

TL;DR

This paper reviews and refines Jensen's result on tilting modules for SL_3(k), applying it to decomposition numbers in symmetric groups and Hecke algebras, and extends findings to quantum groups at roots of unity.

Contribution

It clarifies and completes Jensen's proof for characteristic at least five and extends the results to quantum groups and Hecke algebras at roots of unity.

Findings

Refined Jensen's result on tilting modules for SL_3(k).

Applied results to decomposition numbers for symmetric groups.

Extended results to quantum groups at roots of unity.

Abstract

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.