Orthogonal, symplectic and unitary representations of finite groups

Abstract

Let K be a field, G a finite group, and ρ : G → GL(V ) a linear representation on the finite dimensional K-space V . The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms h : V × V → K which are G-invariant. II. If h is such a form, enumerate the equivalence classes of representations of G into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on G or K under which two orthogonal, symplectic or unitary representations of G are equivalent if and only if they are equivalent as linear representations and their underlying forms are “isotypically” equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) KG-module components of their spaces are equivalent. We assume throughout that the characteristic of K does not divide 2|G|. Solutions to I and II are given when K is a finite or local field, or when K is a global field and the representation is “split”. The results for III are strongest when the degrees of the absolutely irreducible representations of G are odd – for example if G has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index – and, in the case that K is a local or global field, when the representations are split. Let G be a finite group of order g, K a field of characteristic relatively prime to 2g with an involution (possibly the identity) a 7→ ā, and ρ : G → GL(V ) a linear representation of G on the finite dimensional K-vector space V . The group algebraKG has an involution extending that onK and inverting the group elements. Since KG is a semisimple algebra, the involution algebra (KG, ) is a direct sum of simple involution algebras: (KG, ) = (A1, )⊕ (A2, )⊕ · · · ⊕ (Ar, ), where each Ai either is a simple algebra stabilized by the involution or is a “simple hyperbolic algebra” – a direct sum of two simple algebras interchanged by the involution. Now let