Partitions, vertex operator constructions and multi-component KP equations

Abstract

For every partition of a positive integer n in k parts and every point of an infinite Grassmannian we obtain a solution of the k component differential-difference KP hierarchy and a corresponding Baker function. A partition of n also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra gl^ and hence a r function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group Gloo and to express the Baker function in terms of r-functions. The reduction to loop algebras is discussed.

Equations of Hurwitz schemes in the infinite Grassmannian

“…It would be interesting to compare the other hierarchies with those given in [6], which are expressed in terms of pseudodifferential operators and representation theory of infinite dimensional Lie algebras.…”

“…Our approach to the Hurwitz functor is closely related to those given in [2,6,13]. Let us review their approaches in a very concise way.…”

“…This has been studied in works authored by M. R. Adams and M. J. Bergvelt ([2]), M. J. Bergvelt and A. P. E. ten Kroode ( [6]), and Y. Li and M. Mulase ( [13,14]). The soliton hierarchies appearing naturally in this problem are given by the flows defined by the Heisenberg algebras in sl(n, C) (the affine Kac-Moody algebra associated with the loop group of Sl(n, C)).…”

Universality of Rank 6 Plücker relations and Grassmann cone preserving maps

“…Motivation. In the theory of classical integrable systems, it is well-known that the KP hierarchy of soliton equations, written in an appropriate (Hirota) form, is nothing but the Plücker relations for an infinite dimensional Grassmannian in the space of functions [2,15]. Somewhat remarkably, in that setting a single 3-term (i.e., rank 6) quadratic functional equation with parameters suffices to encode the entire hierarchy [6,12,17], and the same 3-term equation, even without the parameters, can characterize Jacobians of curves among all the principally polarized abelian varieties [10].…”

“…The rank of P A,B as a quadratic form on p k n is twice the number |B \ (A ∩ B)| of nonvanishing terms in (2). So the set P(p, n) consists of quadratic forms of every even rank from 6 up to 2 min{p, n − p} + 2, and the Plücker relations in P(p, n) all have rank 6 only when min{p, n − p} = 2.…”