A branching law for the symplectic groups

Abstract

Branching laws for the irreducible tensor representations of the general linear and orthogonal groups are well-known. Furthermore, these laws have a simple form [1]. In the case of the symplectic groups, however, the branching law becomes more complicated and is expressed in terms of a determinant. We derive this result here bybrute force applied to the Weyl character formulas, though it could also have been obtained from a more sophisticated treatment of representation theory contained in some unpublished work of Kostant. The Branching law* Let Vn be an ^-dimensional vector space over the complex field. The symplectic group in n dimensions, Sp(n/2), is the set of all linear transformations ae &(V n), under which a nondegenerate skew-symmetric bilinear form on Vn x V* is invariant, [3]. If is the bilinear form on Vn x Vn and ae ξ?(V n), then