Multiplicities of Schur functions with applications to invariant theory and PI-algebras. (English) Zbl 1056.05141
Recently the authors related to any symmetric function in two variables
\[
f(x,y)=\sum m(\lambda_1,\lambda_2)S_{(\lambda_1,\lambda_2)}(x,y) \in {\mathbb C}[[x,y]],
\]
where \(S_{(\lambda_1,\lambda_2)}(x,y)\) is the Schur function associated with the partition \((\lambda_1,\lambda_2)\), its multiplicity series \(M(f)=\sum m(\lambda_1,\lambda_2) t^{\lambda_1}u^{\lambda_2}\); see V. Drensky and G. K. Genov [C. R. Acad. Bulg. Sci. 55, No. 6, 5–10 (2002; Zbl 1008.16023); J. Algebra 264, No. 2, 496–519 (2003; Zbl 1027.16013)]. In the present paper the authors announce the complete description of the symmetric functions \(f(x,y)\) with the property that \(M(f)\) is a rational function. This happens if and only if \(f\) is of the form \(f(x,y)=(p(x)+p(y))/q(x)q(y)\) for some polynomials \(p(x)\) and \(q(x)\) with coefficients from \({\mathbb C}(xy)\). The authors give also an effective procedure which calculates \(M(f)\). The main results are applied to invariant theory of unipotent linear transformations (or, equivalently, to linear locally nilpotent, or Weitzenböck, derivations) of polynomial algebras, presenting a simple method for calculation of the Hilbert series of the algebra of invariants, to the theory of PI-algebras, and to invariant theory of matrices.
Reviewer: Vesselin Drensky (Sofia)
MSC:
05E05 | Symmetric functions and generalizations |
13A50 | Actions of groups on commutative rings; invariant theory |
15A72 | Vector and tensor algebra, theory of invariants |
16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
16R30 | Trace rings and invariant theory (associative rings and algebras) |