### Abstract

A compound determinant identity for minors of rectangular matrices is established. Given an (s+n-1)×sn matrix A with s blocks of n columns, we consider minors of A by picking up in each block the first consecutive columns specified by weak compositions at most s parts, and prove that the compound determinant of such n×n minors of A is equal to the product of maximal minors of A corresponding to compositions of s+n-1 with s parts. As an application, we obtain Vandermonde type product evaluations of determinants of classical group characters, including Schur functions.

Original language | English |
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Pages (from-to) | 635-654 |

Number of pages | 20 |

Journal | Advances in Applied Mathematics |

Volume | 51 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

- Classical group character
- Compound determinant
- Minor
- Schur function

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Advances in Applied Mathematics*,

*51*(5), 635-654. https://doi.org/10.1016/j.aam.2013.08.001

**A compound determinant identity for rectangular matrices and determinants of Schur functions.** / Ishikawa, Masao; Ito, Masahiko; Okada, Soichi.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 51, no. 5, pp. 635-654. https://doi.org/10.1016/j.aam.2013.08.001

}

TY - JOUR

T1 - A compound determinant identity for rectangular matrices and determinants of Schur functions

AU - Ishikawa, Masao

AU - Ito, Masahiko

AU - Okada, Soichi

PY - 2013/10

Y1 - 2013/10

N2 - A compound determinant identity for minors of rectangular matrices is established. Given an (s+n-1)×sn matrix A with s blocks of n columns, we consider minors of A by picking up in each block the first consecutive columns specified by weak compositions at most s parts, and prove that the compound determinant of such n×n minors of A is equal to the product of maximal minors of A corresponding to compositions of s+n-1 with s parts. As an application, we obtain Vandermonde type product evaluations of determinants of classical group characters, including Schur functions.

AB - A compound determinant identity for minors of rectangular matrices is established. Given an (s+n-1)×sn matrix A with s blocks of n columns, we consider minors of A by picking up in each block the first consecutive columns specified by weak compositions at most s parts, and prove that the compound determinant of such n×n minors of A is equal to the product of maximal minors of A corresponding to compositions of s+n-1 with s parts. As an application, we obtain Vandermonde type product evaluations of determinants of classical group characters, including Schur functions.

KW - Classical group character

KW - Compound determinant

KW - Minor

KW - Schur function

UR - http://www.scopus.com/inward/record.url?scp=84886718637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84886718637&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2013.08.001

DO - 10.1016/j.aam.2013.08.001

M3 - Article

AN - SCOPUS:84886718637

VL - 51

SP - 635

EP - 654

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 5

ER -