arXiv subjects
Publications and source records attributed to Caroline Lassueur.
In a step towards the classification of endotrivial modules for quasi-simple groups, we investigate endotrivial modules for the sporadic simple groups and their covers. A main outcome of our study is the existence of torsion endotrivial modules with dimension greater than one for several sporadic groups with $p$-rank greater than one.
We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.
Motivated by a recent result of Robinson showing that simple endotrivial modules essentially come from quasi-simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of Lie type. Our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods. As one application we prove that the $\ell$-rank of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As a second application we complete the proof that principal blocks of finite simple groups cannot have Loewy length 4, thus answering a question of Koshitani, Külshammer and Sambale. Our results also imply a vanishing result for irreducible characters of special linear and unitary groups.
We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the group structure of $T(G)$ is known for the cases where a Sylow $p$-subgroup $P$ of $G$ is cyclic, semi-dihedral and generalized quaternion. We investigate $T(G)$, and more accurately, its torsion subgroup $TT(G)$ for the case where $P$ is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which $TT(G) = f^{-1}(X(N_{G}(P)))$ holds, where $f^{-1}(X(N_{G}(P)))$ denotes the abelian group consisting of the $kG$-Green correspondents of the one-dimensional $kN_{G}(P)$-modules. We show that the lift to characteristic zero of any indecomposable module in $TT(G)$ affords an irreducible ordinary character. Furthermore, we show that the property of a module in $f^{-1}(X(N_{G}(P)))$ of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.
We investigate simple endotrivial modules of finite quasi-simple groups and classify them in several important cases. This is motivated by a recent result of Robinson showing that simple endotrivial modules of most groups come from quasi-simple groups.