The \(\text{A}_2\) Andrews-Gordon identities and cylindric partitions. (English) Zbl 1518.05009
Summary: 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 \(\text{A}_2\) (or \(\text{A}_2^{(1)}\)) 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 \(\text{A}_{r-1} \) (or \(\text{A}_{r-1}^{(1)}\)) for arbitrary rank \(r\). Our results for \(\text{A}_2\) 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.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
Keywords:
\(\text{A}_{r-1}^{(1)}\) branching functions; character formulas; cylindric partitions; Rogers-Ramanujan identitiesSoftware:
qFunctionsReferences:
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