Schur function expansions of KP $\tau$-functions associated to algebraic curves

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. By 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 $\wp$-functions. The cases of genus-two hyperelliptic and genus-three trigonal curves are detailed as illustrations of the approach developed here.
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