Stable Torsion Pairs in Representation Theory

• Stable torsion pairs are decompositions in module theory that split objects into torsion and torsion-free parts via brick-splitting, refining classical splitting notions.
• They utilize lattice-theoretic properties such as left modularity and trimness to classify torsion classes and reflect the rigidity of module categories.
• Geometric tools like wall-and-chamber structures and Newton polytopes unify combinatorial, algebraic, and geometric stability concepts in the study of these pairs.

A stable torsion pair is a structural notion in module theory, representation theory, and related areas, that encodes how objects (modules, sheaves, or more general categorical objects) can be decomposed into “torsion” and “torsion-free” parts in a way that is compatible with certain homological, categorical, or combinatorial stability phenomena. The modern paper of stable torsion pairs draws on lattice theory, homological algebra, rigidity theory, and geometric combinatorics, providing deep connections between the atomic structure of module categories (as captured by bricks and torsion pairs), the organization of their lattices of subcategories, and the underlying algebraic or geometric source.

1. Definitions: Brick-Splitting and Stability

A brick is an indecomposable module $X$ with $\operatorname{End}_A(X)$ a division algebra. A torsion pair $(\mathcal{T}, \mathcal{F})$ in $\operatorname{mod}A$ is called brick-splitting if every brick $X$ over $A$ lies either in $\mathcal{T}$ or in $\mathcal{F}$ (but not both). This property is a stable analogue of the classical notion of a splitting torsion pair, which requires all indecomposable modules to be so separated. Stability here refers to the persistence of this splitting property under various categorical operations and the reflection of rigid atomic structure (bricks) in the global organization of modules.

Key formal property: For $S \in \mathcal{T}$ and $S' \in \mathcal{F}$ both simple modules (hence bricks), one always has

$\operatorname{Ext}_A^1(S', S) = 0.$

This signals that extensions cannot occur "across the stable wall" separating the torsion and torsion-free bricks.

2. Lattice-Theoretic Characterization of Stable Torsion Pairs

The set $\operatorname{tors}A$ of torsion classes in $\operatorname{mod}A$ forms a complete, semidistributive lattice. In this lattice, the Hasse diagram can be edge-labeled by bricks, via the bijection with completely join- and meet-irreducible elements.

A torsion class $\mathcal{T}$ is brick-splitting if and only if it is left modular in $\operatorname{tors}A$; that is, $x$ is left modular if for all $y \leq z$,

$(y \vee x) \wedge z = y \vee (x \wedge z).$

Thus, the stable structure of brick-splitting torsion pairs is controlled combinatorially by left modularity properties within the lattice of torsion classes.

Characterization Table

Property of $(\mathcal{T}, \mathcal{F})$	Lattice/Theory equivalent	Consequence
Brick-splitting	$\mathcal{T}$ left modular in $\operatorname{tors}A$	Stable separation of bricks
All indecomposables split	$\mathcal{T}$ splitting	Classical (strong) splitting
Each brick labels edge in $[0,\mathcal{T}]$ or $[\mathcal{T},\operatorname{mod}A]$	Direct correspondence	No crossing extensions

3. Brick-Directed Algebras, Cycles, and the Trim Lattice

An algebra $A$ is called brick-directed if there does not exist a cycle of nonzero, noninvertible morphisms between bricks, i.e., bricks are partially ordered by nontrivial morphisms in an acyclic manner. Brick-directedness generalizes the classical notion of representation-directedness.

A trim lattice is a finite lattice that is both extremal (the number of join- and meet-irreducibles equals the length of a maximal chain) and left modular. For a brick-finite $A$, these are equivalent:

• $A$ is brick-directed,
• $\operatorname{tors}A$ is trim,
• $\operatorname{tors}A$ is left modular (all intervals admit left modular maximal chains).

Thus, for brick-finite algebras (those with only finitely many bricks), brick-directedness (stability of the atomic splitting) is precisely reflected in the extremal, modular, and distributive properties of the lattice of torsion classes.

4. Wall-and-Chamber Structure, Stability Functions, and Newton Polytopes

The geometric and combinatorial perspective encodes stable torsion pairs through wall-and-chamber structures on the real Grothendieck group $K_0(\mathrm{proj}A)_\mathbb{R}$, using stability functions. For each brick $X$, define its wall: $$\Theta_X = \{ \theta \in K_0(\mathrm{proj}A)_\mathbb{R} : \theta([X]) = 0 \}.$$ A sequence $\{ \theta_j \}$ (indexed by bricks in a total order) is consistent if, for $i<j$, $\theta_i$ evaluates negatively on $X_j$ and positively for $i>j$. If every brick $X_i$ is $\theta_i$-stable ($\theta_i([X_i])=0$), and such a sequence exists, then $A$ is brick-directed.

This wall-and-chamber structure is dual to the lattice-theoretic properties and is further reflected in the Newton polytope $\mathrm{Newton}(M)$ for $M$ the direct sum of all bricks. Brick-directedness is characterized by the existence of an indivisible increasing path from $0$ to $[M]$ in $\mathrm{Newton}(M)$.

5. Explicit Constructions and Examples

The framework extends to all representation types:

• Dynkin (representation-finite) algebras: All indecomposables are bricks; every torsion pair splitting indecomposables is brick-splitting. The lattice $\operatorname{tors}A$ is distributive and trim.
• Windwheel algebras: Constructed via quivers with loops and relations ensuring the absence of brick-cycles, yet including tame and wild examples. The finite set of bricks can be totally ordered.
• Gluing: By identifying a sink of $A$ with a source of another brick-directed algebra $B$, one constructs new brick-directed algebras with larger rank and potentially wild representation type, and their lattices and stable torsion properties extend accordingly.

The theory thus encompasses the full range of representation-finite, tame, and wild algebras, and provides explicit recipes for constructing algebras with prescribed stable (brick-splitting/brick-directed) torsion-theoretic and lattice properties.

6. Implications and Further Directions

Stable torsion pairs, as embodied in the brick-splitting concept and its lattice-theoretic and geometric avatars, establish a deep correspondence between the atomic, combinatorial, and geometric facets of module categories over finite-dimensional algebras. Notable implications include:

• The classification and combinatorial description of the lattice of torsion classes for classes of algebras previously not accessible by classical splitting theory.
• Sharp invariants for lattice structure (trimness/extremality/left modularity) dictated by atomic stability properties.
• Canonical wall-and-chamber decompositions of $K_0(\mathrm{proj}A)_\mathbb{R}$ linked directly to the asymptotics and combinatorics of module stability and torsion theory.
• Explicit construction of new families of brick-directed algebras, revealing the scope of stable torsion pair theory beyond the classical representation-directed context.

Open directions include the paper of brick-directedness and stable torsion pairs in broader contexts (e.g., in infinite-dimensional settings, derived and triangulated categories); the impact on the structure of $\tau$-tilting finiteness and mutation theory; and applications to the wall-crossing behavior in spaces of stability conditions, as well as the geometry of quiver moduli and Newton polytopes.

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Summary Table of Equivalences and Key Notions

Algebra Property	Lattice Condition	Brick Structure	Wall/Chamber/Polytope
Brick-directed	Torsion lattice trim	No cycles between bricks	Existence of consistent sequence / indivisible path in Newton polytope
Brick-finite	Lattice finite/extremal	Finitely many bricks	Finitely many chambers
Stable torsion pair	Left modular element	Bricks split by $(\mathcal{T},\mathcal{F})$	Wall stability for bricks

This synthesis situates stable torsion pairs as a foundational and unifying concept at the interface of categorical, combinatorial, and geometric representation theory, with profound implications for the understanding of module categories and their invariants (Asai et al., 16 Jun 2025).