An equidistribution conjecture for quotient-closed and submodule-closed subcategories

Comment

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 the number of submodule-closed subcategories of size $i$ for every $i$.

We prove the conjecture for the five smallest and five largest values of $i$, for Nakayama algebras, for algebras with radical square zero, and for representation-directed algebras. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.