Yangians and Capelli Identities

In this article we apply representation theory of Yangians to the classical invariant theory. Let us consider the action of the Lie algebra glN × glM in the space P of polynomial functions on C⊗ C . This action is multiplicity-free, and its irreducible components are parametrized by Young diagrams λ with not more than N,M rows. As a result, the space I of glN × glM invariant differential operators on C⊗ C with polynomial coefficients splits into a direct sum of one-dimensional subspaces parametrized by the diagrams λ . It is easy to describe these subspaces. Let xia with i = 1, . . . ,N and a = 1, . . . ,M be the standard coordinates on the vector space C⊗ C . Let ∂ia be the partial derivation with respect to the coordinate xia . Suppose that the diagram λ consists of n boxes. Let χλ be the irreducible character of the symmetric group Sn parametrized by λ . Then the one-dimensional subspace in I corresponding to λ is spanned by the operator