Représentations Orthogonales et Symplectiques sur un Corps de Caractéristique Différente de 2

Abstract

We give a complete description of primitive linear representations of a p-group over a field F of characteristic ≠2, p, including whether they are of orthogonal or symplectic type. For p odd this is easy and essentially well-known. For p = 2, although the proofs offer no special difficulty the discussion is quite involved, depending on the cyclotomy of F. When F is non-exceptional in the sense of Harris and Segal, primitive representations can be obtained from “generic” ones (Theorems 9.8 and 10.4): this may be considered as the main result of the paper. In passing, we improve an old result of O. Schilling on the division ring of endomorphisms of an irreducible representation of a p-group.

Key Words:

• Finite groups
• Representations
• Quadratic and symplectic

2000 Mathematics Subject Classification:

• Primary 20C15
• Secondary 11E39

Acknowledgments

Notes

Celle-ci est certainement bien connue des experts, mais nous ne sommes pas parvenu à la trouver exposée dans la littérature.

Je remercie le rapporteur de m'avoir signalé ce résultat

^cSi H ne centralisait pas A _i , on aurait H ∩ A _i  ≠ 1, donc A _i  ⊂ H puisque A _i est cyclique d'ordre premier, d'où une contradiction puisque H est supposé abélien.