Pfaffian Formulation of Schur's $Q$-functions

TL;DR

This paper introduces a Pfaffian formula extending Schur's Q-functions to compositions with negative parts, unifying different constructions and enabling new algebraic proofs of identities.

Contribution

It provides a novel Pfaffian formulation for Schur's Q-functions indexed by compositions with negative parts, enhancing theoretical understanding and proof techniques.

Findings

Extended Q-functions to negative parts

Unified Pfaffian with other constructions

Proved identities using algebraic methods

Abstract

We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_\lambda$ to be indexed by compositions $\lambda$ 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.