Schur function expansions of KP τ-functions associated to algebraic curves

J Harnad, Viktor Z Ènol'skii
2011 Russian Mathematical Surveys  
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
more » ... s 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.
doi:10.1070/rm2011v066n04abeh004755 fatcat:w5j55smgj5hxromhhkkzli4m4u