Computing bases for rings of permutation-invariant polynomials (Q1897551)

Let \(R[X_1, \ldots, X_n]\) be the polynomial ring in \(X_1, \ldots, X_n\) where \(R\) is a unitary commutative ring. Given any subgroup \(G\) of the group of permutations of the set \(\{X_1, \ldots, X_n\}\), call \(S : = R[X_1, \ldots, X_n]^G\) the \(R\)-algebra of \(G\)-invariant polynomials. In this paper it is showed how to obtain a basis \(B\) for the \(R\)-algebra \(S\) and it is given an algorithm which allows to represent any element of \(S\) as a linear combination of elements of \(B\) with coefficients in \(R [\sigma_1, \ldots, \sigma_n]\) (where \(\sigma_1, \ldots, \sigma_n\) are the elementary symmetric polynomials). Moreover it is proved that the elements of \(B\) are polynomials of degree bounded by \(\max \{n, n(n - 1)/2\}\). The construction given is independent of the ground ring \(R\).