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

Abstract: 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.

• The paper proposes that quotient-closed and submodule-closed subcategories of a representation-finite algebra are equidistributed by size, and proves the conjecture for algebras with at most nine indecomposables, Nakayama algebras, radical-square-zero algebras, and representation-directed algebras.
• It models quotient closure as a finitary convex geometry whose extreme points are splitting projectives, linking closure structure, functorial finiteness, Auslander–Smalø approximations, and graded subcategory lattices.
• For representation-directed algebras, the paper identifies both counting polynomials with Bruhat intervals in Coxeter groups, while computational and formal verification provides further evidence and highlights that the general representation-finite case remains open.

Overview

This paper, by Haruhisa Enomoto, studies two families of subcategories of the module category of a finite-dimensional algebra: those closed under factor modules (quotient-closed subcategories, also called pretorsion classes) and those closed under submodules. These arise by dropping extension closure from torsion and torsion-free classes, for which the Hom-orthogonal bijection no longer survives. The paper makes two contributions. First, it proposes the quotient–submodule equidistribution conjecture: over a representation-finite algebra $A$ with $N$ indecomposable modules, the number of quotient-closed subcategories containing exactly $i$ indecomposables equals the number of submodule-closed subcategories of size $i$, for every $i$. Second, it develops a structure theory for quotient closure via convex geometries and Coxeter combinatorics. The main theorem proves the conjecture in four cases: algebras with at most nine indecomposable modules, Nakayama algebras, radical-square-zero algebras, and representation-directed algebras (2608.18024).

The convex geometry of quotient closure

The foundational observation is that quotient closure defines a finitary convex geometry on the set $\ind A$ of indecomposable modules. The closure operator is

$\qcl(X)=\ind\Gen(\add X),$

the indecomposable summands of factor modules of finite direct sums from $X$. Finitarity holds because each generated module uses only finitely many generators; anti-exchange follows from Auslander–Smalø's theory of minimal covers: if both $X \in \qcl(Q\cup\{Y\})$ and $Y \in \qcl(Q\cup\{X\})$ held over a closed set $N$0, the splitting projectives of the resulting category would lie in $N$1, contradicting generation of $N$2 or $N$3. Duality gives the analogous result for submodule closure.

The extreme points of this geometry are precisely the splitting projectives—indecomposable modules $N$4 in a quotient-closed subcategory $N$5 such that every surjection onto $N$6 from an object of $N$7 splits. This identification yields several structural consequences: every nonzero quotient-closed subcategory contains a splitting projective; covering relations in the lattice $N$8 differ by one indecomposable, so $N$9 is graded by size; and every quotient-closed subcategory is generated by its splitting projectives.

A notable classification follows: the map $i$0 is a bijection between basic $i$1-minimal modules and functorially finite quotient-closed subcategories, with inverse given by taking splitting projectives. This connects compactness in the convex geometry to functorial finiteness via Auslander–Smalø's approximation theorem.

The Dynkin case and small-size counts

For a Dynkin quiver $i$2 with Weyl group $i$3, the Oppermann–Reiten–Thomas correspondence gives a bijection under which $i$4 corresponds to a quotient-closed subcategory omitting exactly $i$5 indecomposables. Applying this to $i$6 and $i$7 together with duality yields

$i$8

the Poincaré polynomial of $i$9 with exponents $i$0. This establishes the conjecture for hereditary Dynkin algebras.

The paper also develops explicit formulas at the boundary. Size one contributes $i$1 on both sides ($i$2 = number of simples); size $i$3 likewise, since complements correspond to projective and injective modules respectively. At cosize two, the count reduces to $i$4, where $i$5 counts arrows from projectives to nonprojectives in the Auslander–Reiten quiver—and a mesh-correspondence argument shows $i$6 when $i$7 is representation-finite. Notably, this equality can fail already with two omitted modules for representation-infinite algebras: the paper exhibits a specific algebra with $i$8, showing the conjecture genuinely requires representation-finiteness.

Product formulas for Nakayama and radical-square-zero algebras

Nakayama algebras: since indecomposables with fixed top form a chain under quotients, $i$9 decomposes as a product of chains:

$i$0

where $i$1 and $i$2 are composition lengths of projective covers and injective envelopes. A counting argument shows the multisets $i$3 and $i$4 coincide, so the lattices are graded-isomorphic and

$i$5

Radical-square-zero algebras: the separated quiver construction identifies nonsimple indecomposable $i$6-modules with those of the hereditary algebra $i$7 on the separated quiver. Both generating polynomials equal

$i$8

where $i$9 is the Poincaré polynomial of the Coxeter group of the separated graph $\ind A$0 (a disjoint union of Dynkin diagrams by Gabriel's theorem). Equivalently, these are Poincaré polynomials of minimal coset representatives for a parabolic subgroup of type $\ind A$1.

Equidistribution for sizes and cosizes up to four

The uniform boundary result states that $\ind A$2 and $\ind A$3 for all $\ind A$4; since any $\ind A$5 satisfies $\ind A$6 or $\ind A$7 when $\ind A$8, this covers all algebras with at most nine indecomposables.

The proof for sizes two through four uses an annihilator stratification: restriction of scalars decomposes $\ind A$9 into faithful subcategories over factor algebras $\qcl(X)=\ind\Gen(\add X),$0. Ringel's bijection between torsionless modules over $\qcl(X)=\ind\Gen(\add X),$1 and $\qcl(X)=\ind\Gen(\add X),$2 ensures the minimal faithful cores have equal cardinality $\qcl(X)=\ind\Gen(\add X),$3 on both sides. Classification of algebras with $\qcl(X)=\ind\Gen(\add X),$4 (Nakayama, two $\qcl(X)=\ind\Gen(\add X),$5 orientations, and two exceptional loop algebras $\qcl(X)=\ind\Gen(\add X),$6) reduces size four to explicit computations, supported by computer verification.

For cosizes three and four, the key tool is Iyama's theory of $\qcl(X)=\ind\Gen(\add X),$7-categories applied to ideal quotients $\qcl(X)=\ind\Gen(\add X),$8. The paper constructs factor ladders—recursions on the free abelian group on omitted vertices built from almost split sequences—and proves that a complement is quotient-closed precisely when every factor ladder starting at an omitted vertex eventually contains an omitted projective summand. Combined with inclusion–exclusion counting of "projectively rooted" versus "injectively corooted" vertex sets (balanced via the Auslander–Reiten translation), and a finite enumeration of four-vertex patterns (6933 labeled patterns in 329 isomorphism classes, verified computationally), this establishes equality at cosizes three and four.

Bruhat classification for representation-directed algebras

The deepest result concerns representation-directed algebras. Given a Hom-compatible order on $\qcl(X)=\ind\Gen(\add X),$9, the paper builds a simple graph $X$0 whose vertices are $X$1-orbits in the Auslander–Reiten quiver, adjacent when an arrow connects them, and forms the Auslander–Reiten word

$X$2

The main theorem establishes three equivalent characterizations of quotient-closed subcategories: a set $X$3 of omitted positions gives a quotient-closed complement if and only if the mixed coordinates $X$4 are nonnegative, if and only if $X$5 is the lexicographically first reduced subword representing some $X$6 in Bruhat order. Consequently,

$X$7

and the map sending the lexicographically first support to the reverse-lexicographically last support is a size-preserving bijection $X$8.

Two features deserve emphasis. First, unlike the Dynkin case, $X$9 may be infinite—the paper exhibits a radical-square-zero Nakayama algebra whose $X \in \qcl(Q\cup\{Y\})$0 is affine type $X \in \qcl(Q\cup\{Y\})$1—yet the lower interval $X \in \qcl(Q\cup\{Y\})$2 remains finite, with 162 elements there. Second, the proof that reducedness implies positivity of mixed coordinates proceeds through right additive functions on quotient $X \in \qcl(Q\cup\{Y\})$3-categories, identifying coordinate functionals with dimensions of Hom spaces in $X \in \qcl(Q\cup\{Y\})$4. For Dynkin quivers, the construction recovers the Oppermann–Reiten–Thomas correspondence, with $X \in \qcl(Q\cup\{Y\})$5 being the lexicographically first reduced expression of the longest element within $X \in \qcl(Q\cup\{Y\})$6.

Verification and methodology

The paper is accompanied by substantial computational and formal verification: QPA-based checks confirm the conjecture for cluster-tilted algebras of Dynkin type with at most seven simples, Brauer tree algebras with at most five edges, and gentle algebras with at most four simples—including an algebra with 140 indecomposables and over 6.39 billion quotient-closed subcategories whose 141 polynomial coefficients agree on both sides. A Lean 4 formalization exists, and notably, formalization uncovered gaps in the original arguments for sizes three and four that multiple informal reviews had missed; these were subsequently repaired.

Limitations and open questions

The central conjecture remains open in general. The proven cases rely on special structure: lattice decompositions (Nakayama), separated quivers (radical square zero), absence of cycles (representation-directed), or small size. Several limitations are explicit in the paper. The equidistribution phenomenon does not extend to individual lattice structure: even for $X \in \qcl(Q\cup\{Y\})$7 with inward orientation, where the polynomials agree, the lattices $X \in \qcl(Q\cup\{Y\})$8 and $X \in \qcl(Q\cup\{Y\})$9 are not isomorphic, so no permutation-based bijection explains the counts in general. The cosize-two arrow-count equality fails for representation-infinite algebras, indicating that representation-finiteness is essential rather than incidental. Whether the conjecture holds for all representation-finite algebras, whether a canonical size-preserving bijection exists beyond the directed case, and how the Bruhat-interval framework might extend beyond directedness are questions the paper leaves open.

Conclusion

This paper reframes quotient-closed subcategories as closed sets of a finitary convex geometry, derives product formulas and Bruhat-interval classifications in tractable cases, and accumulates strong evidence—both theoretical and computational—for the equidistribution conjecture. The combination of convex-geometric structure theory, $Y \in \qcl(Q\cup\{X\})$0-categorical criteria readable from the Auslander–Reiten quiver, and Coxeter-theoretic classification provides a coherent toolkit that extends the Dynkin picture of Oppermann–Reiten–Thomas substantially, while leaving the general representation-finite case as the natural outstanding problem.