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

O. Foda*, T. A. Welsh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 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proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler.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

Keywords

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

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