Uniqueness of universal dimensions and configurations of points and lines

TL;DR

This paper investigates the uniqueness of universal formulas for quantum dimensions of simple Lie algebras, linking these formulas to geometric configurations like the Pappus-Brianchon-Pascal configuration, and explores non-uniqueness factors.

Contribution

It introduces generic functions that preserve universal dimension formulas and connects these to geometric configurations, revealing non-uniqueness aspects.

Findings

Connection between universal dimension formulas and geometric configurations

Existence of a symmetric non-uniqueness factor involving a realizable configuration

Identification of functions that preserve the structure and values of dimension formulas

Abstract

The problem of uniqueness of universal formulae for (quantum) dimensions of simple Lie algebras is investigated. We present generic functions, which multiplied by a universal (quantum) dimension formula, preserve both its structure and its values at the points from Vogel's table. Connection of some of these functions with geometrical configurations, such as the famous Pappus-Brianchon-Pascal $(9_3)_1$ configuration of points and lines, is established. Particularly, the appropriate realizable configuration $(144_336_{12})$ (yet to be found) will provide a symmetric non-uniqueness factor for any universal dimension formula.