Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

Masao Ishikawa; Masato Wakayama

doi:10.1016/j.jcta.2005.05.008

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

Masao Ishikawa, Masato Wakayama

Research output: Contribution to journal › Article › peer-review

21 Citations (Scopus)

Abstract

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.

Original language	English
Pages (from-to)	113-155
Number of pages	43
Journal	Journal of Combinatorial Theory. Series A
Volume	113
Issue number	1
DOIs	https://doi.org/10.1016/j.jcta.2005.05.008
Publication status	Published - Jan 2006
Externally published	Yes

Keywords

Cauchy identity
Lattice paths method
Lewis Carroll formula
Minor summation formula
Partition
Pfaffian
Plücker relations
Schur function
q-series

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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title = "Applications of minor summation formula III, Pl{\"u}cker relations, lattice paths and Pfaffian identities",

abstract = "The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.",

keywords = "Cauchy identity, Lattice paths method, Lewis Carroll formula, Minor summation formula, Partition, Pfaffian, Pl{\"u}cker relations, Schur function, q-series",

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T1 - Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

AU - Ishikawa, Masao

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AB - The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.

KW - Cauchy identity

KW - Lattice paths method

KW - Lewis Carroll formula

KW - Minor summation formula

KW - Partition

KW - Pfaffian

KW - Plücker relations

KW - Schur function

KW - q-series

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