Hom-Ext Quiver Invariants

• Hom-Ext quiver is defined as a ℤ/2-graded invariant that distinguishes between minimal Hom maps and irreducible Ext classes in exceptional collections.
• It classifies exceptional sets via combinatorial invariants and derived autoequivalences, linking geometric Dehn twists and categorical twist functors.
• Generalizing to superquivers, it connects representation theory with enumerative combinatorics by analyzing partial orders and linear extensions.

A Hom-Ext quiver is a quiver-theoretic invariant that encodes the minimal Hom and Ext interactions among the objects of an exceptional collection, with a ℤ/2-grading structure distinguishing between degree-zero (Hom) and degree-one (Ext) arrows. In recent work, Hom-Ext quivers have been systematized as examples of "superquivers", and their classification, automorphism properties, and applications to the geometry and mutation theory of exceptional sets have been clarified, especially in the case of extended Dynkin type $\widetilde{\mathbb{A}}$ quivers. The Hom-Ext quiver acts as a combinatorial invariant classifying exceptional sets up to derived autoequivalence, realized geometrically via Dehn twists in arc diagram models, and categorically via twist functors in the bounded derived category.

1. Formal Definition and Characterization

A Hom-Ext quiver for an exceptional set $\chi = \{X_1,\ldots,X_m\}$ in a hereditary or finite-dimensional context is defined in two equivalent fashions:

• Endomorphism Quiver Approach: One considers the object $\chi \oplus \Sigma\chi$ in the bounded derived category $\mathcal{D}$, and defines the full quiver $Q^\mathcal{E}$ as the quiver of $$\operatorname{End}_{\mathcal{D}}(\chi \oplus \Sigma\chi)$$, modding out by an equivalence that identifies components according to the shift functor and minimizes non-irreducible morphisms.
• Minimal Morphism Approach: The vertices are labeled by the elements of $\chi$; directed degree-zero arrows are drawn for irreducible morphisms (minimal non-factorizable Hom maps), while degree-one arrows represent extension classes that do not factor through other objects in $\chi$.

This produces a quiver with a $\mathbb{Z}/2$-grading: arrows of degree zero correspond to genuine (irreducible) morphisms, those of degree one to non-split (irreducible) extensions. The path algebra satisfies the rule that any composition of two degree-one arrows is zero. Such a structure is called a "superquiver"; the Hom-Ext quiver is a particular example thereof.

2. Geometric Model and Classification in Type $\widetilde{\mathbb{A}}$

For type $\widetilde{\mathbb{A}}$ (cyclic quivers with acyclic orientation), exceptional collections correspond bijectively to tilings/arc systems in an annulus with marked points on each boundary. In this model, each object $X_i$ corresponds to an oriented arc, and the immediate clockwise relation among arcs determines the directionality of the arrows in the Hom-Ext quiver (cf. Lemma 4.7).

Classification statement: Two exceptional sets in type $\widetilde{\mathbb{A}}$ have isomorphic Hom-Ext quivers if and only if their arc diagrams are related by combinations of Dehn twists along the annulus boundaries. The Hom-Ext quiver thus classifies exceptional sets—up to this twist equivalence—which is also realized categorically by the action of derived autoequivalences generated by twist functors.

Partial order and linear extensions: A natural partial order is induced by the existence of nontrivial paths in the Hom-Ext quiver from $X_i$ to $X_j$. The number of exceptional orderings (total exceptional sequences extending a set) is equal to the number of linear extensions of this partial order (Theorem 4.8), connecting representation theory with classical enumerative combinatorics.

3. Derived Autoequivalence Group—Twist Functors and Dehn Twists

The geometric Dehn twist maps on the annulus correspond to categorical twist functors (Perverse/Seidel-Thomas twists) in the derived category. In type $\widetilde{\mathbb{A}}$:

• Left-tube twist functor $T_\ell$ corresponds to a counterclockwise Dehn twist of the outer boundary.
• Right-tube twist functor $T_r$ corresponds to a clockwise Dehn twist of the inner boundary.

These, together with rescaling functors and automorphisms of the algebra, generate the group of autoequivalences up to shift (Theorem 5.7). The Hom-Ext quiver remains invariant under this group action, making it a complete invariant of the exceptional set up to derived autoequivalence.

4. Superquivers: Generalization and Representation

A superquiver is a quiver $(Q_0, Q_1^{(0)} \sqcup Q_1^{(1)})$ whose arrows are assigned degrees $0$ (Hom) or $1$ (Ext). The path algebra is $\mathbb{Z}/2$-graded, with all paths of total degree $\geq 2$ vanishing. This definition captures the behavior of Hom-Ext quivers and extends beyond—allowing more general configurations.

Exceptional sets over finite acyclic quivers can thus be viewed as $\mathbb{Z}/2$-graded strict representations of superquivers in an ambient triangulated category. One requires that morphisms and extensions respect the grading and that certain compositions vanish according to the superquiver rules. Frozen arrows—connecting two regular modules—are structurally preserved in isomorphisms of superquivers (see Definition 6.2).

5. Enumeration and Partial Orders

Given an exceptional set and its Hom-Ext quiver, the enumeration of exceptional orderings translates to counting linear extensions of the partial order induced by the existence of nonzero Hom/Ext paths. This is generally a #P-complete problem, connecting the subject to classical problems in combinatorics (e.g., counting standard Young tableaux, parking functions, and Tetris tilings). The structure also provides new invariants—potentially representation-theoretic analogues of known enumerative objects.

6. Open Questions and Conjectures

Several natural directions and conjectures are suggested:

• Realizability of Superquivers: Is every combinatorially-possible superquiver (with appropriate frozen arrow data) realized as the Hom-Ext (super)quiver of some exceptional set over a finite acyclic quiver?
• Completeness for Wild/Infinite Types: In more general settings (e.g., wild representation type, non-hereditary algebras), does classification up to Hom-Ext quiver (or superquiver) still reduce the potentially infinite problem of exceptional set classification to a finite question modulo derived autoequivalence?
• Uniqueness of Irreducible Representations: Is every irreducible representation of a superquiver unique up to automorphisms of the ambient triangulated category? This asks for a uniqueness statement generalizing the rigidity known for exceptional sequences in Dynkin types.
• Combinatorial Connections: Are there deeper connections between the combinatorics of linear extensions of partial orders induced by Hom-Ext quivers and other invariants in geometric or representation-theoretic settings?

7. Connection to Existing Quiver and Tilting Theories

In type $\widetilde{\mathbb{A}}$, the Hom-Ext quiver associated to an exceptional set coincides with the tiling algebra of the corresponding geometric model, providing a concrete bridge between categorical, combinatorial, and geometric perspectives. The action of twist functors (realizing Dehn twists) gives a new, explicit presentation of the autoequivalence group in this setting and attaches concrete invariants to the paper of exceptional collections and silting/tilting theory.

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Hom-Ext quivers, as specialized superquivers, thus serve as robust invariants for both classifying and transferring the structure of exceptional sets under derived and geometric symmetries. Their use organizes the typically complex data of exceptional mutations, autoequivalences, and partial orders into a combinatorial and homological framework directly computable from category theory and geometric models. The subject remains active, with key open problems in both the scope of the invariant and in its enumerative and geometric consequences (Igusa et al., 19 Sep 2025).