Cluster DT-Transformation

• Cluster DT-transformation is a specialized automorphism defined by mutation loops that reverse tropical coordinates, ensuring sign stability in cluster algebras.
• It bridges algebraic, combinatorial, and categorical aspects by linking mutation sequences with DT invariants and modeling phenomena like the Auslander–Reiten translation.
• Its quantum version, via quantum dilogarithms, encodes wall-crossing behavior and dynamical invariants such as entropy, fixed points, and spectral stretch factors.

A cluster DT-transformation is a distinguished automorphism acting on the seed or the tropical (and, in the quantized setting, quantum) structure of a cluster algebra, mediating between algebraic, combinatorial, and categorical aspects of cluster dynamics. Originating in the context of Donaldson–Thomas (DT) invariants for 3-Calabi–Yau categories, and realized as special sequences of cluster mutations (often via so-called reddening sequences), cluster DT-transformations serve as a unifying paradigm across the representation theory of quivers, geometric models of moduli spaces, categorification of combinatorics, and dynamical invariants such as entropy and growth rates (Ishibashi et al., 2024, 0811.2435, Bakshi et al., 2024, Weng, 2016, Germain et al., 14 Mar 2025, Choi, 26 Jan 2026, Zhou, 2019, Goncharov et al., 2016, Weng, 2016).

1. Formal Definition and Construction

Let $Q$ be a quiver (typically acyclic, but more generally of finite mutation type) and $\mathcal{A}_Q$, $\mathcal{X}_Q$ the corresponding cluster $\mathcal{A}$- and Poisson $\mathcal{X}$-varieties. A cluster DT-transformation $\tau$ is realized as a specific composition of seed mutations—a "mutation loop"—with the property that in cluster coordinates $(x_i)$ on the positive chamber $C^+=\{x \mid x_i\ge0\ \forall i\}$, the pullback in the tropical semifield satisfies:

$x_i(\tau(w)) = -x_i(w)$

or, more generally, the C-matrix of $\tau$ equals $-I$. In the acyclic case, $\tau$ is canonically specified via a linear extension $\pi$ of the partial order induced by the arrows, producing a mutation loop traversing each mutable vertex exactly once and returning to the original quiver.

On the categorical side, for the path algebra $kQ$, $\tau$ models the Auslander–Reiten translation $\tau_{AR}$ on the module category, and by extension acts as translation on the corresponding derived category via the Coxeter matrix $\Phi$. The quantum and classical (tropical) cluster DT-transformations correspond, through the quantum dilogarithm, to wall-crossing automorphisms encoding DT-invariants in the sense of Kontsevich–Soibelman (0811.2435, Weng, 2016, Goncharov et al., 2016, Ishibashi et al., 2024).

2. Dynamical and Combinatorial Properties

The cluster DT-transformation $\tau$ admits a canonical characterization through "sign stability": consider the tropicalization $X_{\mathrm{trop}}\cong\mathbb{R}^N$ of the cluster variety, and for any seed path $\gamma$ the sign vector of $\gamma$ is eventually stabilized by iterating $\tau$ if and only if certain spectral/combinatorial properties hold.

A crucial invariant is the cluster stretch factor $\lambda_\tau$, the Perron–Frobenius eigenvalue of the matrix linearizing $\tau$ on the sign-stable cone, coinciding with the spectral radius of the Coxeter matrix $\Phi$. The associated entropy formulas then determine the dynamical growth rate:

$$h_{\mathrm{alg}}(\tau) = h_{\mathrm{cat}}(F_\tau) = \log\lambda_\tau$$

For acyclic $Q$, the finite–tame–wild trichotomy is completely characterized by sign stability and the value of $\lambda_\tau$: absence of basic sign stability identifies representation-finite $Q$ (finite type), sign stability with $\lambda_\tau=1$ corresponds to tame type, and $\lambda_\tau>1$ to wild type (Ishibashi et al., 2024).

Furthermore, in finite type cases, cluster DT-transformations are periodic of order $h+2$ (the Coxeter number plus two) (Germain et al., 14 Mar 2025). Every finite type cluster DT admits a unique, totally positive fixed point in the positive real locus, with linearization exponents matching the fundamental degrees of the Weyl group of the associated root system.

3. Quantum Cluster DT and Wall-Crossing

In the quantum setting, the cluster DT-transformation is encoded in the formal automorphism

$$\operatorname{Ad}_{\mathcal{E}_q(Z)}: \hat{T}_{\Gamma,q} \rightarrow \hat{T}_{\Gamma,q}$$

where $T_{\Gamma,q}$ is the quantum torus algebra defined by a skew-symmetric integer pairing, and $\mathcal{E}_q(Z)$ is the quantum DT series determined by stability data and motivic DT-invariants $\Omega_\nu(n\gamma)$. In the cluster case, single-generator factors become quantum dilogarithms and quantum cluster mutations, and in the quasi-classical limit, one recovers the classical cluster DT dynamical system as a piecewise linear automorphism of the tropical torus (0811.2435).

Through the explicit wall-crossing formalism, the quantum cluster DT acts as the unique symplectomorphism sending the positive cone to its negative, implementing wall-crossing in BPS state counting, and matching the Kontsevich–Soibelman theory of 3d Calabi–Yau categories.

4. Realization in Cluster Algebras, Categorification, and Moduli

The existence and explicit form of cluster DT-transformations is established for a wide range of combinatorial and geometric models:

• Acyclic quivers: The DT transformation is always a horizontal mutation loop, and sign stability along with the Coxeter matrix's spectral radius fully determines finiteness and entropy (Ishibashi et al., 2024).
• Grassmannians: The cluster algebra structure of $\operatorname{Gr}(k,n)$ supports a DT transformation realized by a cyclic shift in Plücker coordinates; the associated F-polynomials can be described combinatorially in terms of 3D Young diagrams, reflecting categorification via Cohen–Macaulay module categories and cluster characters (Bakshi et al., 2024, Weng, 2016).
• Double Bruhat cells: Goncharov–Shen proved that the DT on $H\setminus G^{u,v}/H$ coincides (modulo an involution) with the twist map of Fomin–Zelevinsky; this automorphism acts by cluster mutations and underlies duality theorems for canonical theta bases (Weng, 2016).
• Moduli spaces of $G$-local systems: The DT is constructed as the composition of boundary rotation, puncture Weyl reflections, and duality involution, producing the unique automorphism sending basic tropical generators to their negatives (Goncharov et al., 2016).

Moreover, more general birational Weyl group actions on cluster varieties (e.g., $\mathcal{A}_n$ quivers) realize cluster DT-transformations as longest elements, with explicit connection to geodesic flows and invariants of Poisson geometry (Choi, 26 Jan 2026). In the context of log Calabi–Yau surfaces, DT is realized as order-2 Weyl group elements (possibly descending to cluster automorphisms after folding procedures) (Zhou, 2019).

5. Entropy, Fixed Points, and Weyl Group Exponents

For acyclic quivers and their cluster algebras, the algebraic and categorical entropies of the DT transformation directly coincide with $\log\rho(\Phi)$, the logarithm of the spectral radius of the Coxeter matrix. In finite and tame types, this entropy vanishes, as the spectrum lies on the unit circle or the transformation is of finite order.

In finite type, DT transformations are periodic and admit a unique, totally positive fixed point. The linearization of the DT at this fixed point has eigenvalues whose arguments (the "cluster exponents") are in bijection with the degrees of the fundamental Weyl group invariants, providing a direct connection between cluster dynamics and Lie-theoretic structure (Germain et al., 14 Mar 2025).

6. Broader Contexts and Algorithmic Aspects

Cluster DT-transformations, while fundamentally a categorical and algebraic concept, have algorithmic interpretations and appear in related contexts:

• Mutation loops and entropy: General mutation loops can have their sign stability and stretch factors analyzed similarly, and for any mutation loop of finite or tame acyclic quivers, algebraic entropy is always zero (Ishibashi et al., 2024).
• Combinatorial transitions in data clustering: While distinct from the representation-theoretic DT, certain frameworks for constructing transformations between clusterings in combinatorial optimization are also referred to as "cluster DT-transformations" in the sense of staged moves in the configuration space, with minimal circuit walks and transformation metrics (Borgwardt et al., 2019).
• Partition-tree-based multiscale analysis: In high-dimensional data analysis, "cluster DT-transforms" refer to tree-based transforms organizing data according to nested sets, enabling structure-preserving multiscale metrics and clustering, though this is conceptually distinct from the DT in cluster algebra (Mishne et al., 2017).

7. Summary Table: DT-Transformation in Selected Frameworks

Context	Essential Data	Transformation Property	Spectral Invariant
Acyclic quiver algebra	$(Q, \Phi)$	Mutation loop $\tau$, Auslander–Reiten	PF eigenvalue $\lambda_\tau$
Quantum cluster theory	$(\Gamma, \Omega(q))$	Quantum DT (Ad of DT series)	Quantum dilogarithm
Grassmannian	(Ice quiver, $F$-p's)	Cyclic cluster automorphism, F-polys	3D Young diagram sums
Moduli of $G$-local sys	$(\mathcal{X}, w_0, r)$	$C = \star \circ w_0 \circ r$	Weyl group degrees
Partition polytopes	$(X, k)$	Sequential/cyclical moves (combinatorial)	Circuit diameter

In all classical representation-theoretic and cluster algebraic realizations, the cluster DT-transformation is characterized by unique spectral and combinatorial features; entropy, fixed points, and eigenvalue invariants provide the central organizing principles. Its existence, explicit computation, and categorical meaning form the backbone of modern developments in cluster algebra theory, representation theory, and the geometry of moduli spaces (Ishibashi et al., 2024, 0811.2435, Bakshi et al., 2024, Weng, 2016, Germain et al., 14 Mar 2025, Choi, 26 Jan 2026, Goncharov et al., 2016, Zhou, 2019, Weng, 2016).