On the universal Casimir spectrum

We conjecture the connection between sl and so members of universal, in Vogel’s sense, multiplets. The key element is the notion of the vertical componentwise sum [Formula: see text] of Young diagrams. Representations in the decomposition of the power of the adjoint representation of [Formula: see text] algebra can be parameterized by a pair of Young diagrams [Formula: see text] and [Formula: see text] of equal area. We conjecture that within each universal multiplet there exists an associated pair of sl and so representations such that the Young diagram of the so representation is given by [Formula: see text]. This correspondence allows us to derive the universal form of the Casimir eigenvalues on that multiplet. We verify this conjecture for universal decompositions of tensor powers of the adjoint representations up to the fourth power, and for several series of universal representations. Based on this evidence, we propose a set of universal Casimirs for the fifth tensor power of the adjoint representation. We also conjecture that the vertical sum operation is connected with the (dual version of the) folding map of Dynkin diagrams. This connection may explain the intrinsic symmetry of universal formulae under automorphisms of Dynkin diagrams.