Vogel’s universality and refinement

There are quantities in Chern-Simons theory that can be expressed as rational functions of Vogel’s parameters, which accumulate all the dependency on a gauge group. This phenomenon is called algebraic universality in Vogel’s sense. Among these quantities are quantum dimensions, Wilson loop averages (knot invariants) and Chern-Simons partition function. Refinement of Chern-Simons theory means introducing additional parameters, and at the level of symmetric functions it is the transition from Schur symmetric functions to Macdonald polynomials. The question is whether algebraic universality is preserved after refinement. Turns out it survives only for simply-laced root systems. We will compare properties of Schur symmetric functions and Macdonald polynomials associated with different root systems, and discuss algebraic universality of Macdonald dimensions, which are the simplest refined quantity.