Quadratic forms invariant under group actions (Q1186676)

Let \(K\) be a field and \(G\) a finite group. A symmetric, nondegenerate \(K\)-bilinear form \(\beta:V\times V\to K\) on a \(K[G]\)-module \(V\) is said to be a \(G\)-form if \(\beta(gv,gw)=\beta(v,w)\) for all \(v,w\) in \(V\) and \(g\) in \(G\). The Hasse-Witt invariant of the difference of two \(G\)-forms is calculated to obtain explicit criteria for projective isometry of forms. The authors prove that if \(G\) is a \(p\)-group \((p>2)\) and \(V\) is a simple non-trivial \(\mathbb{Q}[G]\)-module, then all positive-definite \(G\)- forms on \(V\) are projectively isometric if and only if \(p\equiv 3\pmod 4\). This generalizes a similar result obtained by \textit{W. Feit} [Linear Multilinear Algebra 3, 25-32 (1975; Zbl 0322.10011)] for cyclic \(p\)- groups. If \(H\) is a subgroup of a group \(G\), then an \(H\)-form on a \(K[H]\)-module \(U\) induces a \(G\)-form on the \(K[G]\)-module \(V=K[G]\otimes_{K[H]}U\). The authors prove that under some assumptions every positive definite \(G\)-form on \(V\) is isometric to an induced form.