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

Abstract: The notion of Kadomtsev-Petviashvili (KP) and BKP $\tau$ functions will be recalled, together with their representations as fermionic expectation values. Schur-type lattices of such KP and BKP $\tau$-functions will be defined, corresponding to a given infinite general linear or orthogonal group element, labelled by partitions and strict partitions respectively. A bilinear expansion expressing elements of these lattices of KP $\tau$-functions as sums over products of pairs of elements of associated lattices of BKP $\tau$-functions will be presented, generalizing earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. Further applications include inhomogeneous polynomial $\tau$-functions of KP and BKP type, with their determinantal and Pfaffian representations.

Refs: 1) F. Balogh, J. Harnad and J. Hurtubise, “Isotropic Grassmannians, Plücker and Cartan maps”, J. Math. Phys. 62, 021701 (2021) 2) J. Harnad and A. Yu. Orlov, “Bilinear expansions of lattices of KP tau-functions in BKP tau-functions: a fermionic approach”, J. Math. Phys. 62, 013508 (2021) 3) J. Harnad and A. Yu. Orlov, “Bilinear expansions of Schur functions in Schur Q-functions: a fermionic approach”, arxiv: 2008.13734 (Proc. Amer. Math. Soc., in press, 2021) 4) J. Harnad and A. Yu. Orlov, “Polynomial KP and BKP τ-functions and correlators”, arXiv:2011.13339, (Ann. H. Poincaré, in press 2021).

HEP - theorymathematical physicsalgebraic geometrycombinatoricsexactly solvable and integrable systems

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