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

Abstract: We derive a bilinear expansion expressing elements of a lattice of KP $\tau$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $\tau$-functions, labelled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. It is deduced using the representations of KP and BKP $\tau$-functions as vacuum expectation values (VEV's) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEV's of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial $\tau$-functions of KP and BKP type.