Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation
Jan E. Grabowski, Matthew Pressland
TL;DR
The paper develops a general, basis-free framework linking cluster categories (in the extriangulated setting) to cluster ensembles, establishing tropical duality between A- and X-sides via indices, coindices, and Grothendieck groups. It defines A- and X-cluster characters and proves they interpolate between categorical data and cluster variables, while also introducing quantum cluster categories through a compatible quantum datum. It shows that mutations of cluster-tilting subcategories categorify the mutation of g- and c-vectors, supported by dualities, sign-coherence, and stabilization results, and extends the theory to infinite rank and loops/2-cycles. The results provide a unified, extensible approach to decategorification, scattering diagrams, and quantum generalizations, with broad applicability to geometrically motivated cluster structures and beyond. Overall, the work offers a robust categorical foundation for cluster ensembles, their tropical data, and their quantisations.
Abstract
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory of $\mathcal{C}$ and the representation theory of $\mathcal{T}$ give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over $\underline{\mathcal{T}}$ to the X-side; here $\underline{\mathcal{T}}$ is the image of $\mathcal{T}$ in the triangulated stable category of $\mathcal{C}$. Moreover, the exchange matrix associated to $\mathcal{T}$ arises from a natural map $p_{\mathcal{T}}\colon\mathrm{K}_0(\operatorname{mod}\underline{\mathcal{T}})\to\mathrm{K}_0(\mathcal{T})$ closely related to taking projective resolutions. Via our approach, we categorify many key identities involving mutation, g-vectors and c-vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements. Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of $p_{\mathcal{T}}$. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß--Leclerc--Schröer.