Cylindric partitions, Wr characters and the Andrews-Gordon-Bressoud identities

O. Foda*, T. A. Welsh

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We study the Andrews-Gordon-Bressoud (AGB) generalisations of the Rogers-Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of sˆlr algebras, the r r,r+d minimal model characters of Wralgebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin's product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor (q;q)-1 , which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud's restricted lattice paths. Extending Bressoud's method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor (q;q)-1 . Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of (q;q)-1 on each side, we obtain the AGB identities.

Original languageEnglish
Article number164004
JournalJournal of Physics. A, Mathematical and theoretical
Volume49
Issue number16
DOIs
Publication statusPublished - 17 Mar 2016

Fingerprint

partitions
Partition
Generating Function
products
Strings
strings
Q-series
Lattice Paths
Algebra
Character
Minimal Model
Ramanujan
Bijection
Partition Function
Limiting
Non-negative
constrictions
algebra
Restriction
Path

Keywords

  • affine and Virasoro characters
  • cylindric partitions
  • Rogers-Ramanujan identities

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

Cite this

Cylindric partitions, Wr characters and the Andrews-Gordon-Bressoud identities. / Foda, O.; Welsh, T. A.

In: Journal of Physics. A, Mathematical and theoretical, Vol. 49, No. 16, 164004, 17.03.2016.

Research output: Contribution to journalArticle

@article{43fb460d45694bf6a45ed9cec89e66bb,
title = "Cylindric partitions, Wr characters and the Andrews-Gordon-Bressoud identities",
abstract = "We study the Andrews-Gordon-Bressoud (AGB) generalisations of the Rogers-Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of sˆlr algebras, the r r,r+d minimal model characters of Wralgebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin's product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor (q;q)-1 ∞, which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud's restricted lattice paths. Extending Bressoud's method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor (q;q)-1 ∞. Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of (q;q)-1 ∞ on each side, we obtain the AGB identities.",
keywords = "affine and Virasoro characters, cylindric partitions, Rogers-Ramanujan identities",
author = "O. Foda and Welsh, {T. A.}",
year = "2016",
month = "3",
day = "17",
doi = "10.1088/1751-8113/49/16/164004",
language = "English",
volume = "49",
journal = "Journal of Physics. A, Mathematical and theoretical",
issn = "1751-8113",
publisher = "IOP Publishing Ltd.",
number = "16",

}

TY - JOUR

T1 - Cylindric partitions, Wr characters and the Andrews-Gordon-Bressoud identities

AU - Foda, O.

AU - Welsh, T. A.

PY - 2016/3/17

Y1 - 2016/3/17

N2 - We study the Andrews-Gordon-Bressoud (AGB) generalisations of the Rogers-Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of sˆlr algebras, the r r,r+d minimal model characters of Wralgebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin's product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor (q;q)-1 ∞, which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud's restricted lattice paths. Extending Bressoud's method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor (q;q)-1 ∞. Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of (q;q)-1 ∞ on each side, we obtain the AGB identities.

AB - We study the Andrews-Gordon-Bressoud (AGB) generalisations of the Rogers-Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of sˆlr algebras, the r r,r+d minimal model characters of Wralgebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin's product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor (q;q)-1 ∞, which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud's restricted lattice paths. Extending Bressoud's method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor (q;q)-1 ∞. Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of (q;q)-1 ∞ on each side, we obtain the AGB identities.

KW - affine and Virasoro characters

KW - cylindric partitions

KW - Rogers-Ramanujan identities

UR - http://www.scopus.com/inward/record.url?scp=84961601685&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/49/16/164004

DO - 10.1088/1751-8113/49/16/164004

M3 - Article

AN - SCOPUS:84961601685

VL - 49

JO - Journal of Physics. A, Mathematical and theoretical

JF - Journal of Physics. A, Mathematical and theoretical

SN - 1751-8113

IS - 16

M1 - 164004

ER -