Completing the A2 Andrews–Schilling–Warnaar Identities

Abstract

AbstractWe study the Andrews–Schilling–Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of ${\operatorname {A}}_2^{(1)}$. These characters have been studied recently by Corteel, Dousse, Foda, Uncu, Warnaar, and Welsh. We prove complete sets of identities for moduli $5$ through $8$ and $10$. The cases of moduli $6$ and $10$ are new. Our proofs use Corteel–Welsh recursions for cylindric partitions and certain relations satisfied by the Andrews–Schilling–Warnaar sum-sides. We further show that at all moduli, a complete set of conjectures may be deduced using certain explicit “seed” conjectures. These seed conjectures are obtained by appropriately truncating our conjectures for the “infinite” level. Additionally, for moduli $3k$, we use an identity of Weierstraß to deduce new sum-product identities starting from the results of Andrews–Schilling–Warnaar.