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Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors

Takeshi Ikeda·Hiro-Fumi Yamada
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Abstract

A family of polynomial τ-functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra \g of type A_1^{(1)}. These τ-functions are given explicitly in terms of Schur functions that correspond to rectangular Young diagrams. It is shown that an arbitrary polynomial τ-function which is an eigenvector of d, the degree operator of \g, is contained in the family. By the construction, any τ-function in the family becomes a Virasoro singular vector. This consideration gives rise to a simple proof of known results on the Fock representation of the Virasoro algebra with c=1.

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Takeshi Ikeda, Hiro-Fumi Yamada. 2002-02-05. Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors. https://arxiv.org/abs/nlin/0202011

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