Linear characters of finite linear groups (Q1291079)

Let \(V\) be a vector space of dimension \(n\) over the field \(\mathbb{F}_q\) of order \(q\), let \(G\) be a subgroup of \(\text{GL}(V)\), and let \(Y\) be a \(G\)-invariant subset of the set \({\mathcal P}(V)\) of \(1\)-spaces of \(V\). The author defines an \(\mathbb{F}_q^*\)-valued linear character \(\delta_Y\) related to the permutation action of \(G\) on \(Y\) and to the associated action of \(G\) on vectors. He shows that special cases of this character include both the determinant when \(G\) is the general linear group and the spinor norm when \(q\) is odd and \(G\) is an orthogonal group. This formulation is particularly useful when one is considering \(G\) as the Galois group of a polynomial.