Universal Quantum Dimensions: γ-Independent Factors

We propose an approach for computing universal, in Vogel’s sense, quantum dimension formulae for universal multiplets whose associated [Formula: see text], [Formula: see text], and [Formula: see text] representations are nonzero. It uses the relation between [Formula: see text] and [Formula: see text] associated representations given by the vertical-sum operation, and the dual relation between [Formula: see text] and [Formula: see text] associated representations given by the horizontal-sum operation on the corresponding Young diagrams of [Formula: see text]. The usual quantum dimensions of these three representations, together with subtleties related to the invariance of universal formulae under automorphisms of the [Formula: see text] Dynkin diagram, allow one to determine the [Formula: see text]-independent factors of a universal quantum dimension (note that [Formula: see text] is the only parameter for classical algebras, depending on their rank). We check this approach on the [Formula: see text]-independent factors of (known) adjoints’ universal quantum dimension, and calculate such factors in the (new) case of the universal multiplet E, in complete agreement with its universal dimension formula. This is another instance in which calculations purely within the classical algebras predict the answers for the exceptional cases, due to the universality structure of the theory of simple Lie algebras.