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Endotrivial Modules for Finite Groups of Lie Type
| Content Provider | Semantic Scholar |
|---|---|
| Author | Mazza, Nadia Nakano, Daniel K. |
| Copyright Year | 2006 |
| Abstract | Let G be a finite group and k be a field of characteristic p > 0. An endotrivial kG-module is a finitely generated kG-module M whose k-endomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided Homk(M, M) ∼= k ⊕ P where P is a projective kGmodule. Now recall that as kG-modules, Homk(M, M) ∼= M∗ ⊗k M where M∗ = Homk(M,k) is the k-dual of M . Hence, the functor “⊗k M” induces an equivalence on the stable module category and the collection of all endotrivial modules makes up a part of the Picard group of all stable equivalences of kG-modules. In particular, equivalence classes of endotrivial modules modulo projective summands form a group that is an essential part of the group of stable self-equivalences. Endotrivial modules were first defined by Dade in [Da]. He demonstrated that for p-groups, the endotrivial modules formed the building blocks of the endo-permutation modules which he proved are the sources for the irreducible modules in finite pnilpotent groups. Later, Puig proved that in fact, the source of a simple module of a finite p-solvable group is an endo-permutation module. Dade also showed that if G is an abelian p-group, then any endotrivial kG-module is the direct sum of a syzygy of the trivial module (Ω(k) for some n, see Section 2 for a definition) and a projective module. More recently, the first author and Thévenaz have given complete classification of the endotrivial modules for p-groups. The group T (G) of endotrivial modules is torsion-free except in the cases that the group G is cyclic, quaternion or semi-dihedral [CaTh2]. The torsion-free rank of T (G) was determined by Alperin [A]. The rank depends on the number of conjugacy classes of maximal elementary abelian p-subgroups of p-rank 2. A complete set of generators for the group of endotrivial modules was given in [CaTh3, Ca2]. The purpose of this paper is to determine the group of endotrivial modules in the defining characteristic for all finite groups of Lie type, including those of twisted type. It is well understood that if G is an arbitrary finite group and M is an endotrivial kG-module, then both the Green correspondent and the source of M are endotrivial modules. For this reason we first consider the endotrivial modules for a Sylow psubgroup U and its normalizer B, a Borel subgroup, of a given finite group G of Lie type. For the unipotent and Borel subgroups we present a complete classification of the endotrivial modules. For the finite groups G of Lie type, T (G) has rank one and is generated by the class of Ω(k) except in cases where the Lie rank is small and the |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://www.maths.lancs.ac.uk/~mazza/cmn.pdf |
| Alternate Webpage(s) | http://eprints.lancs.ac.uk/20807/1/cmn.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | 3,3'-diallyldiethylstilbestrol Class Decision problem Dopamine Dual Irreducibility MANEA gene Maximal set Modulo operation MusicBrainz Picard Semiconductor industry Subgroup A Nepoviruses Torsion (gastropod) Turing completeness Twisted |
| Content Type | Text |
| Resource Type | Article |
