Quantum symmetric functions

We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold (R^m, α) we consider the Poisson orbivariety (R ^m)_n/S_n. The Kontsevich star product on functions on (R^m)_n induces a star product on functions on (R_m)_n/S_n. We provide explicit formulae for the case Heng hooktop sign × Heng hooktop sign/script W sign, where Heng hooktop sign is the Cartan subalgebra of a classical Lie algebra g and script W sign is the Weyl group of Heng hooktop sign. We approach our problem from a fairly general point of view, introducing Polya functors for categories over nonsymmetric Hopf operads.