Geodesic
Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras
Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 137-188.

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Let K be a field of any characteristic. Let the formal power series f(x1, ..., xd) = ∑ αnx1^n1 ··· xd^nd = ∑ 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 whose denominator is a product of binomials of the form 1−x1^a1 ··· xd^ad, we 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 function X M(f;x1, ..., xd ) = ∑ m(λ)x1^λ1 ··· xd^λd, λ = (λ1, ..., λd). M 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.
Keywords: Rational Symmetric Functions, MacMahon Partition Analysis, Hilbert Series, Classical Invariant Theory, Noncommutative Invariant Theory, Algebras with Polynomial Identity, Cocharacter Sequence 
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Benanti, Francesca; Boumova, Silvia; Drensky, Vesselin; K. Genov, Georgi; Koev, Plamen. Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras. Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 137-188. http://geodesic.mathdoc.fr/item/SMJ2_2012_38_1-3_a7/