Irreducible Silting Mutation

• Irreducible silting mutation is a homological operation that replaces one indecomposable summand in a silting object via minimal approximations in exchange triangles.
• It induces a partial order on silting objects and generates a connected silting quiver that reflects key combinatorial and categorical relationships.
• This mutation technique links tilting theory, cluster combinatorics, and torsion pair structures, enabling effective classification in derived and extriangulated categories.

Irreducible silting mutation is a homological operation on silting objects in triangulated and extriangulated categories, typically formulated as the replacement of a single indecomposable summand by another—controlled by approximation and exchange triangles or their extriangulated analogues—so as to produce a new silting object, preserving the silting property. This construction provides both a categorical and combinatorial framework for navigating the space of silting objects, unifying and generalizing classical tilting mutation and closely connecting to cluster theory, support τ-tilting, and the geometry of quivers with potential.

1. Definition and Local Construction

An irreducible silting mutation is the replacement of exactly one indecomposable direct summand in a basic silting object by a new complement, such that the result remains silting. Let $T = T_1 \oplus \cdots \oplus T_n$ be a basic silting object in the bounded derived category $D$ (or a more general Krull–Schmidt triangulated or extriangulated category) associated to a hereditary algebra $H$ or similar algebraic structure.

Fix an indecomposable summand $T_i$, so that $T/T_i$ is an almost complete silting object. The irreducible mutation at $T_i$ proceeds by constructing a triangle

$$M^* \overset{f}{\to} B \overset{g}{\to} M \longrightarrow$$

where $M$ is a complement to $T/T_i$, $g$ is a minimal right $\operatorname{add}(T/T_i)$-approximation, and $f$ is a minimal left $\operatorname{add}(T/T_i)$-approximation. Then $T' = (T/T_i) \oplus M^*$ is again silting, and $M^*$ is the unique other indecomposable complement for $T/T_i$ (up to isomorphism) within this mutation setting (Buan et al., 2010).

In this setup, mutation is called "irreducible" because only one summand is changed, corresponding to a covering relation in the poset or quiver of silting objects (Aihara et al., 2010). The process is controlled homologically via minimal approximations in the triangulated (or extriangulated) structure.

2. Partial Order and the Silting Quiver

Irreducible silting mutation induces a partial order on the set of silting objects, given by: $M \geq N \iff \operatorname{Hom}_D(M, N[>0]) = 0,$ that is, higher extensions from $M$ to $N$ vanish. In this framework, an irreducible left mutation of $M$ is a silting object $N$ such that $M > N$ and no other silting object lies strictly between them. The corresponding quiver—whose vertices are silting objects and arrows are irreducible mutations—is the Hasse quiver for this poset (Aihara et al., 2010).

An important structural property is that, for suitable categories (local, hereditary, or canonical), the action of iterated irreducible silting mutation is transitive: any silting object can be reached from any other by a finite sequence of irreducible mutations. This result associates the connectedness of the silting quiver with deep combinatorial and representation-theoretic properties (Aihara et al., 2010), and is a key finding for representation-finite symmetric algebras as well (Aihara, 2010).

3. Comparison to Other Mutation Theories

Irreducible silting mutation exhibits both similarities and differences with mutation in other settings, such as $m$-cluster tilting objects and exceptional sequences:

• For $m$-cluster tilting objects in $m$-cluster categories $C_m = D/\tau^{-1}[m]$, the corresponding mutation involves replacing a summand and is governed by a finite set (always $m+1$) of complements, ordered by their degree. The combinatorial structure is more rigid here, but the mutation triangles in $C_m$ are directly related to those in $D$, often via a fundamental domain $S_m$ (Buan et al., 2010).
• For exceptional sequences, mutation is more combinatorial and relates to the order of objects in the sequence. Here, mutation is achieved via kernels, cokernels, or universal extensions, rather than via triangles or approximations in the derived category. The operation reorganizes the sequence rather than summands of an object, and the combinatorial tools differ substantively from those of silting mutation (Buan et al., 2010).

Silting mutation in the triangulated or extriangulated context thus interpolates between these worlds: governed by exchange triangles and approximations, with additional flexibility (notably in the infinite or derived setting), and in particular featuring irreducibility when exactly one summand is changed.

4. Structural and Geometric Implications

Irreducible silting mutation provides a powerful tool for constructing and classifying silting objects (and derived equivalences) in various algebraic settings. In categories with Bongartz-type complements, every partial silting object can be completed to a full silting object, and irreducible mutation corresponds to elementary steps in such completions (Aihara, 2010).

Mutation also manifests in geometric models: for example, in the bounded derived category of gentle algebras, irreducible silting mutation is modeled by "flipping" an arc in a graded surface dissection, compatible with the octahedral axiom in the triangulated structure (Chang et al., 2020). More generally, recent developments extend these ideas to extriangulated categories. Here, irreducible silting mutation is encoded by s-triangles in 0-Auslander extriangulated categories, and interval reduction techniques identify "mutation intervals" with silting objects in subquotient categories, controlled by covering relations in the silting poset (Gorsky et al., 2023, Adachi et al., 2023, Pan et al., 24 Jan 2024).

The compatibility between category-theoretic mutation and geometric/combinatorial models (such as in quivers with potential and cluster combinatorics) extends the reach of irreducible silting mutation (Mizuno, 2012, Oppermann, 2015).

5. Connections to Torsion Classes and Module Theory

Irreducible silting mutation is intimately related to the theory of torsion pairs and support τ-tilting modules. Specifically, each basic silting object corresponds to a functorially finite torsion class in the module category, and irreducible mutation translates to minimal changes (covering relations) in the lattice of torsion classes (Hügel et al., 2022, Hügel et al., 2014, Kimura, 2020, Pan et al., 24 Jan 2024). In artinian or length hearts, an irreducible mutation of the corresponding torsion pair is detected by the addition or removal of a single brick (simple object) in the heart, making the mutation theory concrete and combinatorially tractable (Hügel et al., 2022).

Furthermore, the compatibility between silting mutation and universal localization, as well as TTF triples generated by silting modules, allows one to track the effect of irreducible mutation through the spectrum of possible localizations and the associated torsion-theoretic landscape (Marks et al., 2016, Argudin-Monroy et al., 26 Nov 2024).

6. Limitations and Counterexamples

While iterated irreducible silting mutation provides transitivity in many settings, there are explicit examples of finite-dimensional algebras for which the silting quiver decomposes into many components, and not all silting objects can be connected via any sequence of (even non-irreducible) mutations. These obstructions arise from automorphism invariance and the existence of non-invariant spherical modules, highlighting the boundaries of silting "connectedness" (Dugas, 2019).

7. Extensions and Generalizations

Irreducible silting mutation, though classical in the derived module category, has been extended to:

• Extriangulated categories, providing a unified environment for mutation, rigid/silting objects, and their intervals (Gorsky et al., 2023, Pan et al., 24 Jan 2024).
• Geometric models, especially for gentle and surface algebras, where flips of arcs and reduction procedures mirror silting mutation and reduction (Chang et al., 2020).
• Higher Auslander categories, where a d-Auslander extriangulated category corresponds to chains of d irreducible silting mutations between extremal silting objects (Pan et al., 24 Jan 2024).
• Gorenstein silting modules and recollements, further enabling mutation to be tracked through tensor products and glued module categories (Gao et al., 2022).

The robust and flexible nature of irreducible silting mutation—grounded in its local, homological construction and its compatibility with categorical, combinatorial, and geometric frameworks—has rendered it a central tool in the modern paper of derived and representation-theoretic categories.