Macdonald deformation of Vogel's universality and link hyperpolynomials

TL;DR

This paper explores a Macdonald deformation of Vogel's universality in knot theory, deriving universal formulas for hyperpolynomials of Hopf and torus links within the ADE series, highlighting a new layer of universality involving Macdonald dimensions and Littlewood-Richardson coefficients.

Contribution

It introduces a Macdonald deformation framework that extends Vogel's universality to hyperpolynomials of links, revealing universal formulas for ADE series links.

Findings

Universal formulas for hyperpolynomials of Hopf links.

Extension of Vogel's universality to Macdonald dimensions.

Explicit formulas for torus links T[2,2n].

Abstract

Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha,\beta,\gamma$, which are homogeneous coordinates of Vogel's plane. Actually this is true (if at all) only for a piece of representation theory captured by knot/Chern-Simons theory, where some irreducible representations are often undistinguishable and combined into new ``universally-irreducible" entities (uirreps). We consider from this point of view the recently discovered Macdonald deformation of quantum dimensions, for which a kind of universality holds for the ADE series. The claim is that universal are not Macdonald dimensions themselves, but their products with Littlewood-Richardson coefficients, which themselves are functions of $q$ and $t$ in Macdonald theory. These products are precisely what arises in knot/refined Chern-Simons theory. Actually, we consider the simplest decomposition of adjoint square into six uirreps and obtain the universal formulas for hyperpolynomials of the Hopf link and, more generally, of the torus links $T[2,2n]$.