Rational Symmetric Functions MacMahon Partition Analysis Hilbert Series Classical Invariant Theory Noncommutative Invariant Theory Algebras with Polynomial Identity Cocharacter Sequence
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 38, No 1-3, (2012), 137p-188p
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.