Grading the Translation Functors in type A

Abstract

We deal with the representation theory of quantum groups and Hecke algebras at roots of unity. We relate the philosophy of Andersen, Jantzen and Soergel on graded translated functors to the Lascoux, Leclerc and Thibon-algorithm. This goes via the Murphy standard basis theory and the idempotents coming from the Murphy-Jucys operators. Our results lead to a guess on a tilting algorithm outside the lowest $p^2$ alcove, which at least in the $SL_2$-case coincides with Erdmann's results.