Schur function expansions of KP tau functions associated to algebraic curves

TL;DR

This paper reviews the Schur function expansion of KP tau-functions related to algebraic curves, providing explicit formulas for coefficients using derivatives of theta and sigma functions, with detailed examples for genus two and three curves.

Contribution

It introduces explicit expressions for expansion coefficients of KP tau-functions associated with algebraic curves of arbitrary genus, linking them to derivatives of theta and sigma functions.

Findings

Explicit formulas for Plücker coefficients in terms of theta and sigma derivatives

Representation of coefficients as polynomials in Klein's zeta and P functions

Detailed analysis of genus two hyperelliptic and genus three trigonal 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 Pl\"ucker 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.