References

Reduction of permutation-invariant polynomials. A noncommutative case study. (English summary) Inform. and Comput. 175 (2002), no. 2, 158–170. The authors study two problems in noncommutative invariant theory. First, they investigate invariants in the quotient of the free associative algebra in n variables over a commutative ring R by the ideal I generated by the relations Xi(X1 + · · · + Xn) = (X1 + · · · + Xn)Xi, 1 ≤ i ≤ n. Let G be a group of permutations on the variables X1,..., Xn and consider the associated action of G on R〈X1,..., Xn〉/I. An algorithm to express any f ∈ (R〈X1,..., Xn〉/I) G as a polynomial in multilinear G-invariant polynomials is given. Bounds for this algorithm are also stated. Second, the authors begin the analysis of the action of a group G of permutations on the variables X1,..., Xn, on a so-called solvable polynomial ring (these have as an underlying abelian group the commutative polynomial ring R[X1,..., Xn] but with a different noncommutative product of a specific shape). The conditions to extend the action of G on the variables to an action on the solvable polynomial ring are established