Pfaffian formulation of Schur's Q-functions

Abstract

We introduce a Pfaffian formula that extends Schur's Q-functions $Q_{\lambda }$ to be indexed by compositions λ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing $Q_{\lambda }$ into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's Q-functions using only simple algebraic methods.