Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities

We study the Andrews-Schilling-Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of $\mathrm{A}_2^{(1)}$. These characters have been studied recently by various subsets of Corteel, Dousse, Foda, Uncu, Warnaar and Welsh. We prove complete sets of identities for moduli $5$ through $8$ and $10$, in Andrews-Schilling-Warnaar form. The cases of moduli $6$ and $10$ are new. Our methods depend on the Corteel-Welsh recursions governing the cylindric partitions and on certain relations satisfied by the Andrews-Schilling-Warnaar sum-sides. We speculate on the role of the latter in the proofs of higher modulus identities. Further, we provide a complete set of conjectures for modulus $9$. In fact, we show that at any given modulus, a complete set of conjectures may be deduced using a subset of "seed" conjectures. These seed conjectures are obtained by appropriately truncating 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.