Tagged Permissible Arcs in Skew-Gentle Algebras

• Tagged permissible arcs are curves on marked surfaces decorated with combinatorial tags that encode the structure of indecomposable τ‑rigid modules in skew‑gentle algebras.
• Their intersection theory provides a combinatorial framework to compute dimension vectors, Cartan determinants, and to classify support τ‑tilting modules.
• The tagged rotation operation mirrors the Auslander–Reiten translation, linking surface combinatorics directly to module theory and representation classifications.

A tagged permissible arc is a central combinatorial-geometric structure underlying the representation theory of skew-gentle algebras via surface models. On a compact oriented surface $S$ with nonempty boundary and marked points $(S,M,P)$ (boundary points $M$, punctures $P$), tagged permissible arcs are formalized as certain curves endowed with combinatorial decorations (tags) at punctures, subject to compatibility and permissibility constraints set out by a skew-tiling. These objects encode indecomposable $\tau$-rigid modules of algebras arising from partial triangulations, and their intersection theory provides a combinatorial description of homological and categorical data, including dimension vectors, Cartan determinants, and support $\tau$-tilting classification (Deng et al., 16 Nov 2025, He et al., 2020).

1. Geometric and Combinatorial Definitions

A marked surface is $(S, M, P)$ with $S$ a compact oriented surface with $\partial S \neq \emptyset$, $M \subset \partial S$ a finite set of marked boundary points, and $P \subset S \setminus \partial S$ a finite set of punctures.

An arc $\gamma: [0,1] \to S$ satisfies $\gamma(0), \gamma(1) \in M \cup P$, has interior disjoint from $M \cup P \cup \partial S$, and is not homotopic (rel.\ endpoints) to a boundary segment or a point.

A partial triangulation $T$ is a set of pairwise compatible arcs; it is admissible if every puncture is enclosed in a once-punctured monogon cut out by $T$. The regions called tiles are of six possible types (I–VI); a skew-tiling is an admissible partial triangulation where each puncture lies in a type (VI) monogon.

A tagged arc is an arc with each endpoint (if a puncture) tagged either “plain” or “notched,” under rules: (T1) no once-punctured monogon by itself, (T2) boundary endpoints always plain, (T3) both ends of a loop have the same tag.

A tagged permissible arc is a tagged arc whose underlying curve is permissible relative to $T$, meaning it satisfies two local crossing conditions (P1, P2) restricting how arcs may cross tile boundaries and angles (see [(Deng et al., 16 Nov 2025), Figs. 4–5]). Denote the set of all tagged permissible arcs as $\mathbb{PA}(S)$; denote the set of finite multisets of compatible tagged permissible arcs as $\mathscr{R}(S)$.

2. Intersection Numbers and Vectors

For any two tagged arcs $\alpha, \beta$, the intersection number is defined as

$$\mathrm{Int}(\alpha \mid \beta) = \mathrm{Int}^A(\alpha \mid \beta) + \mathrm{Int}^C(\alpha \mid \beta) + \mathrm{Int}^D(\alpha \mid \beta)$$

where $\mathrm{Int}^A$ counts minimal interior crossings, $\mathrm{Int}^C$ is $-1$ if $\alpha,\beta$ form a conjugate pair (i.e., identical curves with tags differing at exactly one puncture), and $\mathrm{Int}^D$ captures mismatches of tags at shared puncture endpoints.

Given a tagged partial triangulation $T^{\bowtie}$ (obtained by replacing each loop at a puncture with two conjugate radii), the intersection vector of $M \in \mathscr{R}(S)$ is

$$\underline{\mathrm{Int}}_{T^{\bowtie}}(M) = \sum_{\gamma \in M} \big(\mathrm{Int}_{T^{\bowtie}}(a \mid \gamma)\big)_{a \in T^{\bowtie}} \in \mathbb{Z}_{\geq 0}^{\,|T^{\bowtie}|}$$

encoding the total intersection numbers of the arcs in $M$ against the arcs of $T^{\bowtie}$ (Deng et al., 16 Nov 2025, He et al., 2020).

3. Uniqueness Theorem and Combinatorial Classification

A fundamental result asserts that, for a skew-tiling $(S, M, P, T)$ with no digons of type (II) or even-gons of type (V), the intersection vector map

$$M \mapsto \underline{\mathrm{Int}}_{T^{\bowtie}}(M)$$

is injective on $\mathscr{R}(S)$—distinct compatible multisets of tagged permissible arcs have distinct intersection vectors. The key steps involve constructing the unfolded tiling $(S^*,M^*,T^*)$, replacing each puncture by a boundary component, and passing to a gentle tiling for which the uniqueness had been established by Fu–Geng. The re-tagging process uniquely recovers $M$ from its image $M^*$ in the unfolded tiling (Deng et al., 16 Nov 2025).

4. Correspondence with Skew-Gentle Algebras and Modules

Each skew-tiling $(S, M, P, T)$ can be encoded as a skew-gentle algebra

$A_T = K Q^T / \langle R^T \rangle$

where the quiver $Q^T$ has vertices indexed by $T$'s arcs, and $R^T$ encodes forbidden compositions. He–Zhou–Zhu show that all skew-gentle algebras arise this way (He et al., 2020).

There is a bijection between $\mathbb{PA}(S)$ and indecomposable $\tau$-rigid modules over $A_T$, with

$\underline{\mathrm{Int}}_{T^{\bowtie}}(\gamma) = \dimv\,M(\gamma)$

where $M(\gamma)$ is the module associated to tagged permissible arc $\gamma$. This extends to multisets and direct sums, identifying intersection vectors with dimension vectors of $\tau$-rigid modules (Deng et al., 16 Nov 2025, He et al., 2020).

5. Even-Cycle Obstructions and Cartan Invariants

The injectivity of the dimension vector map is subject to a fundamental obstruction: If the underlying bound quiver $(Q,I)$ has a minimal oriented cycle of even length where each consecutive pair of arrows composes to zero (a full zero relation), this yields non-isomorphic $\tau$-rigid modules with the same dimension vector, reflected by the Cartan determinant vanishing.

The following conditions are equivalent for a skew-gentle algebra:

1. Different $\tau$-rigid modules have different dimension vectors.
2. The Cartan matrix has nonzero determinant.
3. $(Q,I)$ has no minimal oriented even cycle with full zero relations.
4. The corresponding signed quiver/triple does not have such a cycle.

Precisely, the Cartan determinant is: $$\det(\text{Cartan matrix of }A) = \begin{cases} 0, & ec(Q,I) > 0 \ 2^{\,\#\{\text{odd cycles}\}}, & ec(Q,I) = 0 \end{cases}$$ where $ec(Q,I)$ is the number of minimal even cycles with full zero relations. This situates the geometric theory of tagged permissible arcs as quantifying and explaining the classical “even-cycle” obstruction (Deng et al., 16 Nov 2025).

6. Auslander–Reiten Translation and Tagged Rotation

The Auslander–Reiten translation $\tau$ is realized combinatorially as a tagged rotation: Each tagged permissible arc $(\gamma, K)$ maps to $(p(\gamma), K')$, where $p(\gamma)$ shifts endpoints clockwise to the next marked boundary point, and $K'$ flips the tags at punctures. This bijectively matches the AR-translation on indecomposable modules: $\tau M(\gamma, K) = M(p(\gamma, K))$ Non-projective modules thus yield AR-sequences directly from surface data (He et al., 2020).

7. Support $\tau$-Tilting and Maximal Non-Crossing Laminations

A support $\tau$-tilting module for $A_T$ corresponds to a maximal collection of non-crossing tagged permissible arcs (generalized dissections) in $S$, i.e., a maximally compatible set in $\mathsf{P\!X}(S) \cup T^*$. The bijection is precise: Each such collection yields a basic support $\tau$-tilting pair, and every such pair arises in this fashion. Their cardinality is $|Q_0| + |\operatorname{Sp}| = \dim_K A$. Thus, the surface-combinatorial model fully encodes the support $\tau$-tilting theory of skew-gentle algebras (He et al., 2020).