Rigidity of tilting modules

TL;DR

This paper proves the rigidity of regular tilting modules for quantum groups at roots of unity and explores their Loewy structures, providing explicit examples and extending results to the modular case.

Contribution

It establishes the rigidity of all regular tilting modules for quantum groups at roots of unity and describes their Loewy structures, including explicit examples and modular case extension.

Findings

Regular tilting modules are rigid at roots of unity.

Explicit Loewy structures for Weyl modules in types A2 and B2.

Non-rigid modules appear outside the Jantzen region.

Abstract

Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.