Computing with rational symmetric functions and applications to invariant theory and PI-algebras

Abstract

Let K be a field of any characteristic. Let the formal power seriesbe a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the form 1we use a classical combinatorial method of Elliott of 1903 further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating functionM is a rational function with denominator of a similar form as f . We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.