AI 中文总结
该研究针对任意域上的有限维代数，为τ-刚性对提出平衡记号\(\frac AB\)，简化了τ-倾斜理论公式，方便比较对偶概念，在视觉上有吸引力，还证明在函子有限时相关挠类公式与标准公式在平衡记号下一致。
AI 中文摘要
在任意域\(K\)上的任意有限维代数\(\Lambda\)中，我们为τ-刚性对提出一种新记号：将它们表示为“平衡对”\(\frac AB\)。我们证明在平衡记号下，τ-倾斜理论中的许多公式得以简化。例如，Jasso 范畴及其对偶、与任何τ-倾斜对相关的挠类和无挠类。其他对偶概念，如带符号和余带符号的例外序列，使用平衡记号也更易于比较。平衡记号在视觉上也很吸引人，因为它在遗传情形下与半不变图的壁标签匹配，在一般情形下几乎匹配壁标签。我们还表明，在函子有限的情况下，Igusa 和 Maresca [11]给出的挠类公式在用平衡记号表示时与 Adachi、Iyama 和 Reiten [1]的标准公式一致。
英文摘要
We introduce a new notation for $τ$-rigid pairs called ``balanced pairs''. The notation $\frac AB$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac AB)$ has nice geometric and algebraic implications: It defines ``lower'' and ``upper'' chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.
Comments37 pages, 10 figures. The idea behind this paper was presented at the 2021 Workshop on Cluster Algebras and Related Topics at the Morningside Center for Mathematics at CAS in Beijing, August 2021. v2: added details and theorems. Submitted to Axioms Journal
