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

We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) τ-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP τ-functions, labeled by strict partitions. This generalizes earlier results relating determin...

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Bibliographic Details
Published in:Journal of mathematical physics 2021-01, Vol.62 (1)
Main Authors: Harnad, J., Orlov, A. Yu
Format: Article
Language:English
Subjects:
Functions (mathematics)
Lattices
Mathematical analysis
Matrix methods
Operators (mathematics)
Partitions
Physics
Polynomials
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Summary:We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) τ-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP τ-functions, labeled 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 τ-functions as vacuum expectation values (VEVs) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by the 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 VEVs of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial τ-functions of KP and BKP type.
ISSN:0022-2488
1089-7658
DOI:10.1063/5.0032525