The ${\mathrm{A}}_2$ Andrews-Gordon identities and cylindric partitions

Abstract

Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers-Ramanujan-type identities, we obtain the ${\mathrm{A}}_2$ (or ${\mathrm{A}}_2^{(1)}$) analogues of the celebrated Andrews-Gordon identities. We further prove $q$-series identities that correspond to the infinite-level limit of the Andrews-Gordon identities for ${\mathrm{A}}_{r-1}$ (or ${\mathrm{A}}_{r-1}^{(1)}$) for arbitrary rank $r$. Our results for ${\mathrm{A}}_2$ also lead to conjectural, manifestly positive, combinatorial formulas for the $2$-variable generating function of cylindric partitions of rank $3$ and level $d$, such that $d$ is not a multiple of $3$.

arXiv comment46 pages, minor typos corrected, to appear in Transactions of the AMS, Series B

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• Lepowsky's and Wakimoto's product formulas for the affine Lie algebras $C_l^{(1)}$