### 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.

Original language | English |
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Pages (from-to) | 147-156 |

Number of pages | 10 |

Journal | Letters in Mathematical Physics |

Volume | 60 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2002 |

### Keywords

- Nonlinear Schrödinger equation
- Schur functions
- Virasoro algebra

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Ikeda, T., & Yamada, H. F. (2002). Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors.

*Letters in Mathematical Physics*,*60*(2), 147-156. https://doi.org/10.1023/A:1016167008456