Symmetric power sum expansions of the eigenvalues of generalised Casimir operators of semi-simple Lie groups

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Abstract
A formula due to Okubo (1975) for the eigenvalues of generalised Casimir operators of semi-simple Lie groups is used to drive an explicit expression for these eigenvalues. Full use is made of the Weyl symmetry group and is shown that this expression may be cast in the form of a symmetric power sum expansion. Expansions are derived for operators of order p constructed using the defining representation of each simple Lie group for all p<or=8. The results are in accord with the known facts regarding a complete set of algebraically independent operators and yield algebraic relations amongst those which are not independent. The expansions for the orthogonal and symplectic groups are a distinct improvement upon those obtained earlier, whilst those for the exceptional groups are the first of their kind.