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Symmetric Functions and the KP and BKP Hierarchies

P. Jarvis
C. Yung
Journal / Anthology

Journal of Physics A
Year: 1993
Volume: 26
Page range: 5905-5922

We study the KP hierarchy through its relationship with S-functions. Using results from the classical theory of symmetric functions, the Plücker equations for the hierarchy are derived from the tau function bilinear identity and are given in terms of composite S-functions. Their connection to the Hirota bilinear form of the hierarchy is clarified. A novel combinatorial proof is given of the fact that Schur polynomials solve the KP hierarchy. We show how the analysis can be carried through for the BKP hierarchy in a completely parallel fashion, with the S-functions replaced by Schur Q-functions.

*Mathematics > Calculus and Analysis > Differential Equations