Endotrivial modules for the sporadic simple groups and their covers. (English) Zbl 1348.20011
From the introduction: 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 1 for several sporadic groups with \(p\)-rank greater than 1.
Let \(G\) be a finite group and \(k\) a field of prime characteristic \(p\) dividing the order of \(G\). A \(kG\)-module \(V\) is called endotrivial if \(V\otimes V^*\cong k\oplus Q\), with \(Q\) a projective \(kG\)-module. The tensor product over \(k\) induces a group structure on the set of isomorphism classes of indecomposable endotrivial \(kG\)-modules, called the group of endotrivial modules and denoted \(T(G)\). This group is finitely generated and it is of particular interest in modular representation theory as it forms an important part of the Picard group of self-equivalences of the stable category of finitely generated \(kG\)-modules. In particular the self-equivalences of Morita type are induced by tensoring with endotrivial modules.
Our paper is organised as follows. In Sections 2 and 3, we sum up useful results on endotrivial modules and the sporadic groups. In Section 4, we determine the structure of \(T(G)\) in characteristic 2. For odd characteristics, in Section 5 we determine the torsion-free rank of \(T(G)\), and in Section 6 the torsion subgroup \(TT(G)\) when we have obtained its full structure. We collect the results in Section 7 in Table 3 if \(p^2\mid |G|\) and in Table 5 if the Sylow \(p\)-subgroups are cyclic.
Let \(G\) be a finite group and \(k\) a field of prime characteristic \(p\) dividing the order of \(G\). A \(kG\)-module \(V\) is called endotrivial if \(V\otimes V^*\cong k\oplus Q\), with \(Q\) a projective \(kG\)-module. The tensor product over \(k\) induces a group structure on the set of isomorphism classes of indecomposable endotrivial \(kG\)-modules, called the group of endotrivial modules and denoted \(T(G)\). This group is finitely generated and it is of particular interest in modular representation theory as it forms an important part of the Picard group of self-equivalences of the stable category of finitely generated \(kG\)-modules. In particular the self-equivalences of Morita type are induced by tensoring with endotrivial modules.
Our paper is organised as follows. In Sections 2 and 3, we sum up useful results on endotrivial modules and the sporadic groups. In Section 4, we determine the structure of \(T(G)\) in characteristic 2. For odd characteristics, in Section 5 we determine the torsion-free rank of \(T(G)\), and in Section 6 the torsion subgroup \(TT(G)\) when we have obtained its full structure. We collect the results in Section 7 in Table 3 if \(p^2\mid |G|\) and in Table 5 if the Sylow \(p\)-subgroups are cyclic.
MSC:
| 20C20 | Modular representations and characters |
| 20C34 | Representations of sporadic groups |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
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