Orthogonal, symplectic and unitary representations of finite groups.

Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are: \par {\bf I.} {\em Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant.} \par {\bf II.} {\em If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group).} \par {\bf III.} {\em 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.} \par 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. \par We assume throughout that the characteristic of $K$ does not divide $2|G|$. \par Solutions to {\bf I} and {\bf 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 {\bf 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. [ABSTRACT FROM AUTHOR]