砖链滤过的有限性与增长
Finiteness and growth of brick chain filtrations
- Parakeet Inc.（鹦鹉株式会社）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有限维代数上Ringel砖链滤过的有限性，证明固定模的最大子模取值有限，给出维数d的界2^{d^2}，并构造Kronecker代数上滤过计数指数增长的砖族，回答Ringel的两个问题。
AI中文摘要：
我们研究有限维代数上的Ringel砖链滤过：这类滤过的因子被单个砖的拷贝所滤过，并排序使得较早砖到较晚砖的态射消失。首先，利用子模簇，我们证明固定模的属于某个挠类的最大子模随类变化仅取有限多个值，回答了Pavón的问题。由此推出砖链滤过的有限性以及维数d的模的界2^{d^2}。对于τ-倾斜有限代数，我们将其数量界定为由简单合成重数决定的多项式系数。最后，我们在三箭Kronecker代数上构造砖族，其滤过计数随合成长度平方呈指数增长。我们利用子模计数的Littlewood–Richardson公式和钩长公式证明此增长。特别地，计数最终超过合成长度的阶乘。这些结果回答了Ringel的两个问题。
英文摘要：
We study Ringel's brick chain filtrations over finite-dimensional algebras: filtrations whose factors are filtered by copies of individual bricks, ordered so that morphisms from earlier bricks to later ones vanish. First, using submodule varieties, we prove that the largest submodule of a fixed module belonging to a torsion class takes only finitely many values as the class varies, answering Pavón's question. We deduce finiteness of brick chain filtrations and the bound $2^{d^2}$ for modules of dimension $d$. For $τ$-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth using a Littlewood--Richardson formula for submodule counts and the hook-length formula. In particular, the counts eventually exceed the factorial of composition length. These results answer Ringel's two questions.
