Orthogonal, symplectic and unitary representations of finite groups (Q2750919)

Let \(K\) be a field, \(G\) a finite group, \(\rho\) a linear representation on the finite dimensional \(K\)-space \(V\), and the characteristic of \(K\) does not divide \(2|G|\).NEWLINENEWLINENEWLINEThe following problems are considered: (i) determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms \(h\colon V\times V\to K\) 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. 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. Problem (iii) is considered for many particular cases. Furthermore, if \(K\) is a global field, it is given a proof of the Hasse principle for equivalent representations of finite groups over \(K\).