On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>↔</mml:mo> <mml:mo>−</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> self-duality of su(N) theories

We generalize N ↔ − N duality of dimension formulae of SU ( N ) representations on a (class of the) representations with N -dependent Young diagrams, including adjoint representation. Duality is extended on the eigenvalues of Casimir operator for these representations. We discuss the consequences for the hypothesis of universal, in Vogel’s sense, decomposition of powers of adjoint representation into Casimir subspaces.