On the universal Casimir spectrum

Abstract

We conjecture the connection between $su$ and $so$ members of universal, in Vogel's sense, multiplets. The key element is the notion of the {\it vertical componentwise sum} $\oplus_v$ of Young diagrams. Representations in the decomposition of the power of the adjoint representation of $su(N)$ algebra can be parameterized by a couple of $N$-independent Young diagrams $\lambda$ and $\tau$, with equal area. We assume that the $so(N)$ member of the universal (Casimir) multiplet of a given $su(N)$ representation is the $so$ representation with $\lambda \oplus_v \tau$ Young diagram. This allows one to obtain the universal form of the Casimir eigenvalue on that multiplet. Conjecture is checked for all known cases: universal decompositions of powers of adjoint up to fourth, and series of universal representations. On this basis we suggest the set of universal Casimirs for fifth power of adjoint. We also conjecture that vertical sum operation is a kind of the (dual version of the) folding map of Dynkin diagrams. This will hopefully explain the intrinsic symmetry of universal formulae with respect to the automorphisms of Dynkin diagrams.