Let \( G \) be a finite group. It is not hard to see that for any representation \( \rho: G \to \mathrm{GL}(V) \) for \( V \) a real vector space, there exists a \( G \)-invariant bilinear form \( \beta \) on \( V \), i.e., a non-degenerate bilinear form such that \( \beta(\rho(g) v, \rho(g) w)=\beta(v,w) \) for all \( g \in G, v,w \in V \). If \( \rho \) is "orthogonally stable" (so is a sum of even-dimensional irreducible real representations), then the square class of the determinant of the Gram matrix for any basis (the "orthogonal determinant") does not depend on the choice of \( \beta \), giving us interesting invariants of our group \( G \). Richard Parker conjectured that these orthogonal determinants are always "odd", for any finite group. We will see that the conjecture holds for the symmetric groups, as well as the general linear groups \( \mathrm{GL}(q) \) for \( q \) a power of an odd prime. In the discussion, important concepts like (standard) Young tableaux and Iwahori--Hecke algebras will come up. This talk has the additional purpose of giving a small introduction (with many examples) into the representation theory of finite groups. As such, no previous knowledge in that area will be assumed.