Quantum dimensions in the universal branching rule for the symmetric square of the adjoint
Year: 2026
Type: preprint
Abstract: For a simple complex Lie algebra π€ with Vogel parameters (Ξ±, Ξ², Ξ³), let ΞΈ be the highest root, π°π©β^ΞΈ its π°π©β-subalgebra and π€ββ² the centralizer of π°π©β^ΞΈ. A companion paper decomposes each Vogel piece Yβ(Ξ½) of the symmetric square of the adjoint under π°π©β^ΞΈ β π€ββ² by sixteen labels, rational functions on Vogel's plane that are added class by class. We lift this rule to quantum dimensions. Thirteen labels become products of q-numbers of linear forms, built from the point of π€ and the shifted points of its two blocks. The other three are fixed by the quantum dimension of Yβ(Ξ½); they are not such products, but their sum has a closed form. We prove, by a classification-based and computer-assisted argument, that for every simple π€ other than π°π©β and π°π¬β the quantum dimension of every class, in the principal specialization of π€, is the sum of its q-labels; a block of Dynkin index k enters at q^(1/k). The ingredients are the splitting Ο_π€ β Ο_π€ββ² = (t β 1)ΞΈ/2 of Landsberg and Manivel, with t the dual Coxeter number, a root-string lemma and a closed product for the Cartan components V(Ξ» + ΞΈ_X), a three-term decomposition of V β π€β²_X, V = π€β, for each block π€β²_X, and a proof, series by series, of Mkrtchyan's quantum dimension formulas for Yβ(Ξ²) and Yβ(Ξ³) and of Westbury's formula for the adjoint. We record an exact check of 687 classes on 30 algebras, an independent recomputation on 13 algebras, and a prediction of the classes of Aβ, Aβ and Dβ from the five exceptional algebras. The scripts in anc/ use exact arithmetic and reproduce every computation cited in the paper; each log ends with "checks passed: True".
Author: Guideng, Eric
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225834326