Efficient generation of the ring of invariants (Q1911393)

By using the Binet-Minc formula in the theory of permanents, the author proves David Richman's theorem: Let \(G\) be a finite group on \(A:= R(a_1, \dots, a_r)\), where \(R\) is any commutative ring with \(1/|G|!\in R\) (\(|G|\) denotes the order of \(G\)). Then the ring of invariants \(A^G\) is generated over \(R\) by \(\sum_{\sigma\in G} \sigma (a_1^{\alpha_1} a_2^{\alpha_2} \dots a_r^{\alpha_r})\), where \(\alpha_1+ \dots +\alpha_r\leq |G|\). In the course of discussion, applications of permanents to other problems related to invariants, e.g., the properties of symmetric polynomials, are also given.