A generalization of Weyl's identity for \(D_{n}\) (Q705252)

From Weyl's identity for the root system of type D comes an explicit linear combination of Schur functions. In this paper this identity is generalised and the Schur function expansion is found, but this is not all. It is arguable that the most important contribution of this paper is to give to the subjects of mathematics and physics a clear, concise and accessible account of the connection between the Fermionic Fock space, which arises in Dirac's theory of electrons and positrons, and the algebra of symmetric functions, traditionally associated with the representation theory of the symmetric group. After clearly laying out the necessary notation and definitions the existence of a Fermionic Fock space is proved via an explicit construction before showing that all others are isomorphic to it. From here operators are introduced and their commutation relations are proved using the Wick-Lieb identity and a Schur lemma, both of which are verified here. All these facets are then drawn together and connected to the algebra of symmetric functions before being used to prove the generalisation of Weyl's identity.