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 series f(x1, . . . , xd) = ∑ αnx n1 1 · · ·x nd d = ∑ 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−x1 1 · · ·x ad d , 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)x1 1 · · ·x λd d , 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 theory. Introduction Let K be a field of any characteristic and let K[[X]]d be the subalgebra of the symmetric functions in the algebra of formal power series K[[X]] = K[[x1, . . . , xd]] in the set of variables X = {x1, . . . , xd}. We study series f(X) ∈ K[[X]]d which can be presented as rational functions with denominators products of binomials of the form 1 − X = 1 − x1 1 · · ·x ad d . Following Berele [15], we call such functions nice rational symmetric functions. Nice rational symmetric functions appear in many places in mathematics. For the examples inspired our project, let K be of characteristic 0. If W is a polynomial module of the general linear group GLd = GLd(K), then its GLd-character is a symmetric polynomial which gives W the structure of a graded vector space. Hence the Hilbert (or Poincare) series of the symmetric algebra K[W ] is a nice rational symmetric function. Nice rational symmetric functions appear as Hilbert series in classical invariant theory. For example, this holds for the Hilbert series of the pure trace algebra 2010 Mathematics Subject Classification. 05A15; 05E05; 05E10; 13A50; 15A72; 16R10; 16R30; 20G05.