COMPUTING WITH RATIONAL SYMMETRIC FUNCTIONS AND APPLICATIONS TO INVARIANT THEORY AND PI-ALGEBRAS

To Yuri Bahturin on the occasion of his 65th birthday Abstract. Let K be a field of any characteristic. Let the formal power series f(x1,..., xd) = αnx n1 1 · · ·xndd = m(λ)Sλ(x1,..., xd), αn,m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function with denominator which is a product of binomials of the form 1−xa11 · · ·xadd, we use a classical combinatorial method of Elliott in 1903 further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating function M(f;x1,..., xd) = m(λ)xλ11 · · ·xλdd, λ = (λ1,..., λd), which is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, in classical invariant theory, algebras with polynomial identities and noncommutative invariant the-ory