AI 中文总结
该研究针对与平方积箭图$A_2\square A_2$相关的凯勒$q$-双对数恒等式，通过生成函数、带色分拆双射、四色弧代数三种方法，以迭代Durfee分解为核心完成了组合与代数证明，补充了该恒等式的新证明途径。
AI 中文摘要
经典的Durfee平方论证为量子双对数的五边形恒等式提供了组合证明。正如五边形恒等式与$A_2$箭图相关联，凯勒（Keller）恒等式则与平方积箭图$A_2\square A_2$相关联。此前凯勒恒等式的证明使用了簇范畴或快速衰减等变上同调中的谱序列。我们给出凯勒恒等式的三种证明：生成函数证明、带色分拆上的显式保权双射，以及使用四色弧代数的标准单项式证明。它们的共同机制是迭代Durfee分解：两种可能的配对分别给出水平和垂直分解，而第三个二元步骤则解释耦合因子。这些构造为Durfee平方论证提供了“超势类似物”。
英文摘要
The classical Durfee-square argument gives a combinatorial proof of the pentagon identity for the quantum dilogarithm. Just as the pentagon identity is associated with the $A_2$ quiver, Keller's identity is associated with the square-product quiver $A_2\square A_2$. Previous proofs of Keller's identity use cluster categories or spectral sequences in rapid-decay equivariant cohomology. We give three proofs of Keller's identity: a generating-function proof, an explicit weight-preserving bijection on colored partitions, and a standard-monomial proof using a four-colored arc algebra. Their common mechanism is an iterated Durfee decomposition: two possible pairings give horizontal and vertical decompositions, while a third binary step accounts for the coupling factor. These constructions provide a ``superpotential analogue'' of the Durfee-square argument.
