A generalization of Weyl's identity for Dn

Abstract

The expansion in Schur functions of the product ∏ i< j (1− x i x j ) is well known. It is more or less equivalent to Weyl's identity for root-systems of type D n . In this paper we obtain the expansion in Schur functions of the product ∏ i< j (1−∑ ℓ>0 a ℓ x i x j ( x i ℓ− x j ℓ)/( x i − x j )), thus generalizing Weyl's identity. We obtain this result by systematic calculation in fermionic Fock space. But familiarity with the latter is not a prerequisite for reading this paper. We develop from scratch the modest amount of theory that we need in elementary and purely algebraic fashion, taking pains to integrate the theory with classical symmetric function theory. The tools developed in this paper ought to have many further applications, e.g., to random matrix theory and to computational abelian function theory.