Universal Quantum Dimensions: I. $γ$-Independent Factors

TL;DR

The paper advances a method to derive universal quantum dimension formulae by exploiting vertical-sum relations between $sl$ and $so$ representations and horizontal-sum relations between $sl$ and $sp$ representations, anchored by the Weyl-dimension framework and automorphism invariance. By comparing the $sl$, $so$, and $sp$ quantum dimensions for known universal multiplets, it isolates the $\gamma$-independent factors and demonstrates the reconstruction on the adjoint and a new $E$-type representation. The approach offers a compact, cross-validated route to universal dimensions, highlighting hidden universality that extends predictions from classical to exceptional algebras. Future work is planned to extend to $\gamma$-dependent factors and other universal multiplets, potentially clarifying the full structure of universal dimension formulae.

Abstract

We propose a method for computing universal (in Vogel's sense) quantum dimension formulae for universal multiplets whose associated $sl$, $so$, and $sp$ representations are nonzero. The method uses the relation between $sl$ and $so$ representations given by the vertical-sum operation, and the dual relation between $sl$ and $sp$ representations given by the horizontal-sum operation on the corresponding Young diagrams. The usual quantum dimensions of these three representations, together with subtleties related to the invariance of universal formulae under automorphisms of the $sl$ Dynkin diagram, allow one to determine the $γ$-independent factors of a universal quantum dimension (note that $γ$ is the only parameter for classical algebras, depending on their rank). Using this approach, we compute the $γ$-independent factors for (known) adjoints' universal quantum dimension, and also obtain such a factor in one new case. We discuss how to extend this approach to the $γ$-dependent factors in the quantum dimension formulae, and other issues. This is another instance in which calculations purely within the classical algebras predict the answers for the exceptional cases, due to the hidden universality structure of the theory of simple Lie algebras.