Polynomial KP and BKP $\tau$-functions and correlators

Abstract

Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.

26 pages. References updated. Notations for $Q_{\alpha }$ rendered more consistent. Statement and proof of Propositions 4.1 and 4.2 revised. Eq. (4.6) corrected. Typos corrected

References in corpus (4)

• Polynomial Tau-functions of the KP, BKP, and the s-Component KP Hierarchies
• Bilinear expansions of lattices of KP $\tau$-functions in BKP $\tau$-functions: a fermionic approach
• Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach
• Finite Dimensional KP τ-functions I. Finite Grassmannians

Cited by in corpus (6)

• The generalized Giambelli formula and polynomial KP and CKP tau-functions
• Tau functions, infinite Grassmannians and lattice recurrences
• Extended Schur's $Q$-functions and the full Kostant--Toda hierarchy on the Lie algebra of type $D$
• Unitary matrix integrals, symmetric polynomials, and long-range random walks
• Notes about KP/BKP correspondence
• Fredholm Pfaffian $\tau$-functions for orthogonal isospectral and isomonodromic systems