2015/01/02 by Caroline Lassueur, Lassueur, Caroline, Gunter Malle +1
Mathematics · #20C20 (Primary) #20C30 #20C33 #20C34 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20C20 #msc:20C30 #msc:20C33 #msc:20C34
paper · pdf · doi:10.48550/arxiv.1501.00400
29 pages
arxiv created 2015/01/02 · arxiv updated 2015/01/05
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 ℓ-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.