  The descent set of an oscillating (or up-down) tableau is introduced. This
descent set plays the same role in the representation theory of the symplectic
groups as the descent set of a standard tableau plays in the representation
theory of the general linear groups. In particular, we show that the descent
set is preserved by Sundaram's correspondence. This gives a direct
combinatorial interpretation of the branching rules for the defining
representations of the symplectic groups; equivalently, for the Frobenius
character of the action of a symmetric group on an isotypic subspace in a
tensor power of the defining representation of a symplectic group.
