Vogel's universality and Macdonald dimensions

TL;DR

This paper explores the universality of Macdonald dimensions within algebraic structures, providing a universal formula that unifies these quantities across simply laced root systems and introduces mixed dimensions.

Contribution

It introduces a universal formula for Macdonald dimensions and duals, extending Vogel's universality to refined algebraic quantities associated with simple Lie algebras.

Findings

Unified formula for Macdonald dimensions across simply laced root systems

Definition of dual and mixed Macdonald dimensions

Extension of Vogel's universality to refined quantities

Abstract

We discuss algebraic universality in the sense of P. Vogel for the simplest refined quantity, the Macdonald dimensions. The main known source of universal quantities is given by Chern-Simons theory. Refinement of Chern-Simons theory means introducing additional parameters. At the level of symmetric functions, the refinement is the transition from the Schur functions to the Macdonald polynomials. We consider the Macdonald polynomials associated with the simple Lie algebras, define Macdonald dimensions and dual Macdonald dimensions, and present a universal formula for them that unifies these quantities for algebras associated with simply laced root systems. We also consider mixed Macdonald dimensions that depend on two different root systems.