Let \(K\) be a field and \(V\) an \(n\)-dimensional vector space over \(K\). Let \(\{x_1, \dots, x_n\}\) be a basis for \(V\). Let \(S^n\) denote the symmetric group acting on a basis of \(V\). Let \(\text{GL}_m (K)\) denote the group of \(m \times m\) invertible matrices over \(K\), \(\mathbb{P}(V)\) the projective space associated with \(V\), \(S^{(k)} V\) the \(k\)th symmetric power of \(V\), and \(\wedge^{(k)} V\) the \(k\)th exterior power of \(V\). Suppose that some representation \(S_n \to \text{GL} (V) \simeq \text{GL}_m (K)\) is given. The authors show that the fixed fields \(K(S^{(k)} V)^{S_n}\), \(K (\mathbb{P} (S^{(k)} V))^{S_n}\), and \(K(\wedge^{(k)} V)^{S_n}\) are rational over \(K\). If \(K\) has characteristic \(p > 0\), then the authors show that \(K(S^{(p^m)} V_0)^{S_n}\) and \(K(\mathbb{P} (S^{(p^m)}V_0))^{S_n}\) are rational over \(K\), where \(V_0\) is the quotient space of \(V\) by the subspace \(K \cdot (x_1 + \cdots + x_n)\). The authors also consider some other cases.