Tau functions, infinite Grassmannians and lattice recurrences

Abstract

The addition formulae for KP τ -functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the τ -functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon-Pemantle) recurrence.

The generalized Giambelli formula and polynomial KP and CKP tau-functions

“…We want to study n-reductions of the CKP hierarchy, which means that we restrict c ∞ to the Lie algebra c ∞ ∩ ĝl n , where ĝl n = gl n (C[t, t −1 ] ⊕ K is the subalgebra of a ∞ , consisting of all n-periodic matrices g = (g ij ) i,j∈ 1 2 +Z , i.e. g i+n,j+n = g ij , together with K. This intersection is equal to the affine Lie algebra ŝp n if n is even, and to the twisted affine Lie algebra ĝl (2) n if n is odd (see [9, p 977]). Let G be an element in the group G n , corresponding to this affine Lie algebra.…”

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

“…It was Sato [93,94] and Segal and Wilson [96] who pioneered the connection between Fredholm Grassmannians and integrable systems. Recently there has also been a resurgence in this direction as well, see for example, Mulase [76], Dupré et al [36,37,38], Kasman [63,64], Hamanaka and Toda [59], Cafasso [18], Cafasso and Wu [19] and Arthamonov et al [7] (also see Beck et al [10,11,12] and Doikoi et al [31,32]). Related to this is the well-studied connection between the Korteweg-de Vries hierarchy, the intersection theory of Deligne-Mumford moduli space and the a string equation in two-dimensional gravity; see for example Witten [105,106] and Cafasso and Wu [19].…”