Canonical bases for cluster algebras

By Mark Gross, P. Hacking, S. Keel, M. Kontsevich. 2014

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In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H0(Y,OY ) extending to a basis of H0(U,OU ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SLr, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/60e28184fbedf965ea6426ca336e426044637085

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Citations in the corpus

Positivity for quantum cluster algebras (2016, methodology): “Specialising the quantum cluster positivity conjecture at q = 1 we obtain the classical positivity conjecture of Fomin and Zelevinsky [FZ02], recently proved by Lee and Schiffler [LS15] in the skew-symmetric case, and then in the case of geometric type by Gross, Hacking, Keel and Kontsevich [GHKK14]”
Descendant log Gromov-Witten invariants for toric varieties and tropical curves (2016, background): “In [Man], the first author showed that theta functions on cluster varieties as in [GHKK14] can indeed be expressed in terms of descendant tropical curve counts.”
A survey on maximal green sequences (2019, methodology): “veloped in [35, 36]. Alternative proofs of the theorem were given in [99] (via the cluster category) and in [95] (via Donaldson–Thomas theory). An important generalization to valued quivers is proved [59]. As shown in [96], the theorem is central in the theory of cluster algebras. The c-vector αi P ”
The twist for positroid varieties (2016, background): “onical divisor, making it an example of an aﬃne log Calabi-Yau variety with maximal boundary, in the sense of [GHK15]. This may be the source of the cluster structure, according to the perspective of [GHKK14]. THE TWIST FOR POSITROID VARIETIES 41 1 2 4 3 5 6 7 8 4678 1678 1268 1238 2348 2345 2456 45”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “llknown almost only in quiver case. The author hopes that future research will allow the usage of the existence theorem in symmetrizable cases. The recent paper by Gross, Hacking, Keel and Kontsevich [GHKK14] provided anotherpositive basis ofcommutative cluster algebrasparametrized by tropical point”
Positroid varieties and cluster algebras (2019, background): “ny consequences for the structure of open positroid varieties, see e.g. the introduction of [SSBW19]. For instance, the existence of a green-to-red sequence [FS18], together with the constructions of [GHKK18] endow C[ v;w] with a basis of theta functions with positive structure constants. Additiona”
Wall and chamber structure for finite-dimensional algebras (2018, background): “Later, Gross, Hacking, Keel and Kontsevich studied in [21] the so-called cluster scattering diagram, proving several conjectures on cluster algebras.”