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proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler.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.
|Journal||Journal of Physics. A, Mathematical and theoretical|
|Publication status||Published - 17 Mar 2016|
- affine and Virasoro characters
- cylindric partitions
- Rogers-Ramanujan identities