### 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proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler.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 sˆl_{r} algebras, the _{r} ^{r,r+d} minimal model characters of W_{r}algebras, 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 language | English |
---|---|

Article number | 164004 |

Journal | Journal of Physics. A, Mathematical and theoretical |

Volume | 49 |

Issue number | 16 |

DOIs | |

Publication status | Published - 17 Mar 2016 |

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### 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

_{r}characters and the Andrews-Gordon-Bressoud identities.

*Journal of Physics. A, Mathematical and theoretical*,

*49*(16), [164004]. https://doi.org/10.1088/1751-8113/49/16/164004

**Cylindric partitions, W _{r} characters and the Andrews-Gordon-Bressoud identities.** / Foda, O.; Welsh, T. A.

Research output: Contribution to journal › Article

_{r}characters and the Andrews-Gordon-Bressoud identities',

*Journal of Physics. A, Mathematical and theoretical*, vol. 49, no. 16, 164004. https://doi.org/10.1088/1751-8113/49/16/164004

_{r}characters and the Andrews-Gordon-Bressoud identities. Journal of Physics. A, Mathematical and theoretical. 2016 Mar 17;49(16). 164004. https://doi.org/10.1088/1751-8113/49/16/164004

}

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 -