Reduction of Permutation-Invariant Polynomials: A Noncommutative Case Study

Let R be a commutative ring with 1, let RX₁,…,X_n/I be the polynomial algebra in the n≥4 noncommuting variables X₁,…,X_n over R modulo the set of commutator relations I={(X₁+···+X_n)*X_i=X_i*(X₁+···+X_n)|1≤i≤n}. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X₁,…,X_n, and let RX₁,…,X_n/I^G be the R-algebra of G-invariant polynomials in RX₁,…,X_n/I. The first part of this paper is about an algorithm, which computes a representation for any fRX₁,…,X_n/I^G as a polynomial in multilinear G-invariant polynomials, i.e., the maximal variable degree of the generators of RX₁,…,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 RX₁,…,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.