### Abstract

This paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald's book. We refer to these identities as the "Littlewood-type formulas." In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy-Binet formula. The Cauchy-Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain.

Original language | English |
---|---|

Pages (from-to) | 480-525 |

Number of pages | 46 |

Journal | Journal of Algebra |

Volume | 208 |

Issue number | 2 |

Publication status | Published - Oct 15 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*208*(2), 480-525.

**New Schur function series.** / Ishikawa, Masao; Wakayama, Masato.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 208, no. 2, pp. 480-525.

}

TY - JOUR

T1 - New Schur function series

AU - Ishikawa, Masao

AU - Wakayama, Masato

PY - 1998/10/15

Y1 - 1998/10/15

N2 - This paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald's book. We refer to these identities as the "Littlewood-type formulas." In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy-Binet formula. The Cauchy-Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain.

AB - This paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald's book. We refer to these identities as the "Littlewood-type formulas." In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy-Binet formula. The Cauchy-Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain.

UR - http://www.scopus.com/inward/record.url?scp=0346502382&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346502382&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0346502382

VL - 208

SP - 480

EP - 525

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -