Generalizations of Cauchy's determinant and Schur's Pfaffian

Masao Ishikawa, Soichi Okada, Hiroyuki Tagawa, Jiang Zeng

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det(1/(xi+yj)) and Schur's Pfaffian Pf((xj-xi)/(xj+xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood-Richardson coefficients involving a rectangular partition.

Original languageEnglish
Pages (from-to)251-287
Number of pages37
JournalAdvances in Applied Mathematics
Volume36
Issue number3
DOIs
Publication statusPublished - Mar 2006
Externally publishedYes

Fingerprint

Pfaffian
Cauchy
Determinant
Vandermonde determinant
Littlewood-Richardson Coefficients
Partition
Generalise
Generalization

Keywords

  • Cauchy's Determinant
  • Pfaffian
  • Plücker relations
  • Schur functions

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Generalizations of Cauchy's determinant and Schur's Pfaffian. / Ishikawa, Masao; Okada, Soichi; Tagawa, Hiroyuki; Zeng, Jiang.

In: Advances in Applied Mathematics, Vol. 36, No. 3, 03.2006, p. 251-287.

Research output: Contribution to journalArticle

Ishikawa, Masao ; Okada, Soichi ; Tagawa, Hiroyuki ; Zeng, Jiang. / Generalizations of Cauchy's determinant and Schur's Pfaffian. In: Advances in Applied Mathematics. 2006 ; Vol. 36, No. 3. pp. 251-287.
@article{ea632abba0ae49fdb8a109ea88929475,
title = "Generalizations of Cauchy's determinant and Schur's Pfaffian",
abstract = "We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det(1/(xi+yj)) and Schur's Pfaffian Pf((xj-xi)/(xj+xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood-Richardson coefficients involving a rectangular partition.",
keywords = "Cauchy's Determinant, Pfaffian, Pl{\"u}cker relations, Schur functions",
author = "Masao Ishikawa and Soichi Okada and Hiroyuki Tagawa and Jiang Zeng",
year = "2006",
month = "3",
doi = "10.1016/j.aam.2005.07.001",
language = "English",
volume = "36",
pages = "251--287",
journal = "Advances in Applied Mathematics",
issn = "0196-8858",
publisher = "Academic Press Inc.",
number = "3",

}

TY - JOUR

T1 - Generalizations of Cauchy's determinant and Schur's Pfaffian

AU - Ishikawa, Masao

AU - Okada, Soichi

AU - Tagawa, Hiroyuki

AU - Zeng, Jiang

PY - 2006/3

Y1 - 2006/3

N2 - We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det(1/(xi+yj)) and Schur's Pfaffian Pf((xj-xi)/(xj+xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood-Richardson coefficients involving a rectangular partition.

AB - We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det(1/(xi+yj)) and Schur's Pfaffian Pf((xj-xi)/(xj+xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood-Richardson coefficients involving a rectangular partition.

KW - Cauchy's Determinant

KW - Pfaffian

KW - Plücker relations

KW - Schur functions

UR - http://www.scopus.com/inward/record.url?scp=33645021456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645021456&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2005.07.001

DO - 10.1016/j.aam.2005.07.001

M3 - Article

VL - 36

SP - 251

EP - 287

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 3

ER -