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An identity is derived expressing Schur functions as sums over products ofpairs of Schur $Q$-functions, generalizing previously known special cases. Thisis shown to follow from their representations as vacuum expectation values(VEV's) of products of either charged or neutral fermionic creation andannihilation operators, Wick's theorem and a factorization identity for VEV'sof products of two mutually anticommuting sets of neutral fermionic operators.

Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach