商闭与子模闭子范畴的等分布猜想
An equidistribution conjecture for quotient-closed and submodule-closed subcategories
AI总结：
该研究针对有限维代数的模范畴中对商或子模封闭的子范畴，提出商-子模等分布猜想并证明四类情形，还拓展了商闭子范畴的分类方法。
AI中文摘要：
我们研究有限维代数的模范畴中对商或子模封闭的子范畴，提出商-子模等分布猜想：在表示有限代数上，对每个i，大小为i的商闭子范畴的数量等于大小为i的子模闭子范畴的数量，其中大小指子范畴中不可分解模的个数。我们证明该猜想的以下情形：(1) i的五个最小值与五个最大值，因此所有至多有9个不可分解模的代数；(2) Nakayama代数；(3) 根平方为零的代数；(4) 表示定向代数。在最后一种情形中，我们通过由Auslander-Reiten箭图构造的Coxeter群中的下Bruhat区间对商闭子范畴分类，扩展了Oppermann-Reiten-Thomas对Dynkin箭图的分类。我们还证明商闭包定义了一种有限凸几何，并通过Gen极小模对函子有限的商闭子范畴分类。
英文摘要：
We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient--submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander--Reiten quiver, extending the classification of Oppermann--Reiten--Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.
