A universal branching rule for the symmetric square of the adjoint under highest-root breaking
Year: 2026
Type: preprint
Abstract: For a simple complex Lie algebra š¤ with Vogel parameters (Ī±, Ī², Ī³), the symmetric square of the adjoint decomposes as SĀ²š¤ = ā ā Yā(Ī±) ā Yā(Ī²) ā Yā(Ī³). Let Īø be the highest root, š°š©ā^Īø its š°š©ā-subalgebra, š¤āā² its centralizer, and H = š°š©ā^Īø ā š¤āā² (the quaternionic symmetric pair). We decompose each Yā(Ī½) under H by a single rule on Vogel's plane. An isotypic component is labeled by Ī½, the š°š©ā^Īø-spin and the eigenvalue of the Casimir of š¤āā². Sixteen labels suffice, each with a Casimir value linear in the parameters and a dimension rational in them; the dimension of a class is the sum of its labels, added as rational functions and then evaluated. We prove the rule for every simple š¤ other than š°š©ā and š°š¬ā, assuming Vogel's parametrization of the Casimir spectrum on SĀ² for š¤ and for the simple algebras in its chain of blocks; the proof also uses an index condition and the location of the point Ī² = Ī³, both read off the classification, and dim V(2Īø) = 300 for š°š¬ā. Vogel's dimension formula for the pieces is not assumed; it follows by induction on the rank. The key steps are a Casimir identity on the odd part of SĀ²š¤, which reduces spin 1/2 to a Vandermonde moment problem, and a three-moment computation at spin 0. Corollaries include the analogous rule for ĪĀ²š¤, which needs the parametrization for š¤ alone, and the number of H-invariants in each piece. We record verification on 31 algebras, a computer-assisted integrality theorem on six lines of the plane with a conjecture for the whole plane, and an identity of level-one central charges that holds on the whole plane.The scripts in anc/ use exact rational arithmetic and reproduce every computation in the paper, including the computer-assisted proof of the integrality theorem (anc/ubr_int.py).
Author: Guideng, Eric
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225831671