We introduce a Pfaffian formula that extends Schur's Q-functions Q_λ 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_λ 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.
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Graf et al. (2024) studied this question.
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