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We derive a bilinear expansion expressing elements of a lattice of KP$\tau$-functions, labelled by partitions, as a sum over products of pairs ofelements of an associated lattice of BKP $\tau$-functions, labelled by strictpartitions. This generalizes earlier results relating determinants andPfaffians of minors of skew symmetric matrices, with applications to Schurfunctions and Schur $Q$-functions. It is deduced using the representations ofKP and BKP $\tau$-functions as vacuum expectation values (VEV's) of products offermionic operators of charged and neutral type, respectively. The lattice isgenerated by insertion of products of pairs of charged creation andannihilation operators. The result follows from expanding the product as a sumof monomials in the neutral fermionic generators and applying a factorizationtheorem for VEV's of products of operators in the mutually commutingsubalgebras. Applications include the case of inhomogeneous polynomial$\tau$-functions of KP and BKP type.

Bilinear expansions of lattices of KP $\tau$-functions in BKP $\tau$-functions: a fermionic approach