论文标题
完成$ \ mathrm {a} _2 $ andrews-shilling-warnaar身份
Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities
论文作者
论文摘要
我们研究了$ \ mathrm {a} _2 _2^{（1）} $的标准特征（即，可集成，最高权重）模块的主要特征的Andrews-Shilling-warnaar sum端。这些角色最近由Corteel，Dousse，Foda，Uncu，Warnaar和Welsh的各种子集研究。我们证明了Moduli $ 5 $至8美元和$ 10 $的完整身份，以Andrews-Shilling-Warnaar表格。 Moduli $ 6 $和$ 10 $的情况是新的。我们的方法取决于管理圆柱形分区的Corteel-Welsh递归以及安德鲁斯 - 刺耳的 - 瓦纳尔总成所满足的某些关系。我们推测后者在较高模量身份的证据中的作用。此外，我们为模量$ 9 $提供了一套完整的猜想。实际上，我们表明，在任何给定的模量下，可以使用“种子”猜想的子集推导一组猜想。这些种子的猜想是通过适当截断“无限”水平的猜想获得的。此外，对于Moduli $ 3K $，我们使用Weierstrass的身份来推断新的总产品身份，从Andrews-Shilling-Warnaar的结果开始。
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.
