Quotient–Submodule Equidistribution Conjecture

Establish that for every representation-finite finite-dimensional algebra A over a field, the size-generating polynomials of quotient-closed and submodule-closed subcategories coincide; equivalently, prove that the numbers of quotient-closed and submodule-closed subcategories containing exactly i isomorphism classes of indecomposable modules are equal for every i.

Background

The paper studies two families of additive subcategories of the module category of a finite-dimensional algebra: quotient-closed subcategories, also called pretorsion classes, and submodule-closed subcategories. For a representation-finite algebra, the size of such a subcategory is the number of indecomposable modules it contains, and the corresponding size-generating polynomials record the number of subcategories of each size.

The conjecture asserts coefficientwise equality between the two generating polynomials for every representation-finite finite-dimensional algebra. The paper proves the conjecture for several substantial classes, including Nakayama algebras, algebras with radical square zero, and representation-directed algebras, as well as for the smallest and largest four sizes and cosizes. However, the paper explicitly states that the conjecture is not proved in full, so the general case remains unresolved.