Schur function expansions of KP tau functions associated to algebraic curves

Abstract

The Schur function expansion of Sato-Segal-Wilson KP tau-functions is reviewed. The case of tau-functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Plücker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann theta function or Klein sigma function along the KP flow directions. Using the fundamental bi-differential, it is shown how the coefficients can be expressed as polynomials in terms of Klein's higher genus generalizations of Weierstrass' zeta and P functions. The cases of genus two hyperelliptic and genus three trigonal curves are detailed as illustrations of the approach developed here.

48 pages

References in corpus (1)

• Abelian Functions for Trigonal Curves of Genus Three

Cited by in corpus (14)

• Free fermions and tau-functions
• Classical tau-function for quantum spin chains
• The master T-operator for the Gaudin model and the KP hierarchy
• On higher genus Weierstrass sigma-function
• Tau functions as Widom constants
• Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions
• Sato Grassmannian and Degenerate Sigma Function
• The generalized Giambelli formula and polynomial KP and CKP tau-functions
• On the Expansion Coefficients of Tau-function of the BKP Hierarchy
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Finite Dimensional KP τ-functions I. Finite Grassmannians
• Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation
• Abelian function fields on Jacobian varieties
• Quantum Gaudin model and classical KP hierarchy