### 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 2002 |

### Fingerprint

### Keywords

- Nonlinear Schrödinger equation
- Schur functions
- Virasoro algebra

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Letters in Mathematical Physics*,

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

**Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors.** / Ikeda, Takeshi; Yamada, Hiro Fumi.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 60, no. 2, pp. 147-156. https://doi.org/10.1023/A:1016167008456

}

TY - JOUR

T1 - Polynomial τ-functions of the NLS-Toda hierarchy and the Virasoro singular vectors

AU - Ikeda, Takeshi

AU - Yamada, Hiro Fumi

PY - 2002/5

Y1 - 2002/5

N2 - 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.

AB - 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.

KW - Nonlinear Schrödinger equation

KW - Schur functions

KW - Virasoro algebra

UR - http://www.scopus.com/inward/record.url?scp=0040670080&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040670080&partnerID=8YFLogxK

U2 - 10.1023/A:1016167008456

DO - 10.1023/A:1016167008456

M3 - Article

AN - SCOPUS:0040670080

VL - 60

SP - 147

EP - 156

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 2

ER -