Reduction of permutation-invariant polynomials

  • Authors:

  • Manfred Göbel;Heinz Kredel

  • Affiliations:

  • Dettenbachstraße, Neukirchen vorm Wald, Germany;Univ. Mannheim, Mannheim, Germany

  • Venue:
  • Information and Computation
  • Year:
  • 2002

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Abstract

Let R be a commutative ring with 1, let RX1,..., Xn/I be the polynomial algebra in the n4 noncommuting variables X1,..., Xn over R modulo the set of commutator relations I={(X1++Xn)Xi=Xi(X1++Xn)|1in}. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X1,..., Xn, and let RX1,..., Xn/IG be the R-algebra of G-invariant polynomials in RX1,..., Xn/I. The first part of this paper is about an algorithm, which computes a representation for any fRX1,..., Xn/IG as a polynomial in multilinear G-invariant polynomials, i.e., the maximal variable degree of the generators of RX1,..., Xn/IG 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,..., Xn/IG 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.