  Enhanced global non-abelian symmetries at zero coupling in Yang Mills theory
play an important role in diagonalising the two-point functions of multi-matrix
operators. Generalised Casimirs constructed from the iterated commutator action
of these enhanced symmetries resolve all the multiplicity labels of the bases
of matrix operators which diagonalise the two-point function. For the case of U
(N) gauge theory with a single complex matrix in the adjoint of the gauge group
we have a U(N)^{\times 4} global symmetry of the scaling operator at zero
coupling. Different choices of commuting sets of Casimirs, for the case of a
complex matrix, lead to the restricted Schur basis previously studied in
connection with string excitations of giant gravitons and the Brauer basis
studied in connection with brane-anti-brane systems. More generally these
remarks can be extended to the diagonalisation for any global symmetry group G.
Schur-Weyl duality plays a central role in connecting the enhanced symmetries
and the diagonal bases.
