Reduction of Permutation-Invariant Polynomials

Abstract

Let R be a commutative ring with 1, let R 〈 X 1 ,…, X n 〉 / I be the polynomial algebra in the n ≥4 noncommuting variables X 1 ,…, X n over R modulo the set of commutator relations I ={( X 1 +···+ X n )∗ X i = X i ∗( X 1 +···+ X n )|1≤ i ≤ n }. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X 1 ,…, X n , and let R 〈 X 1 ,…, X n 〉 / I G be the R -algebra of G -invariant polynomials in R 〈 X 1 ,…, X n 〉 / I . The first part of this paper is about an algorithm, which computes a representation for any f ∈ R 〈 X 1 ,…, X n 〉 / I G as a polynomial in multilinear G -invariant polynomials, i.e., the maximal variable degree of the generators of R 〈 X 1 ,…, X n 〉 / I G is at most 1. The algorithm works for any ring R and for any permutation group G . In addition, we present a bound for the number of necessary generators for the representation of all G -invariant polynomials in R 〈 X 1 ,…, X n 〉 / I G with a total degree of at most d . The second part contains a first but promising analysis of G -invariant polynomials of solvable polynomial rings.