Cylindric partitions, W_r 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proof of Borodin's product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler.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 affine sl_r algebras, the M^{r, r+d}_r minimal model characters of W_r algebras, and the r-string abaci generating functions, as well as the relationships between them, providing simple proofs for each. We then set r=2, and use 2-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin's product expression for the generating functions of the 2-cylindric partitions with infinitely-long parts, to obtain the product sides of the AGB identities, times a factor (q; q)_{\infty}^{-1}, which is the generating function of ordinary partitions. Next, we obtain a bijection from the 2-cylindric partitions, via 2-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 2-cylindric partitions which contains a factor (q; q)_{\infty}^{-1}. Equating the product and sum expressions of the same 2-cylindric partitions, and canceling a factor of (q; q)_{\infty}^{-1} on each side, we obtain the AGB identities.