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Canonical bases for cluster algebras


Authors: Mark Gross, Paul Hacking, Sean Keel and Maxim Kontsevich
Journal: J. Amer. Math. Soc. 31 (2018), 497-608
MSC (2010): Primary 13F60; Secondary 14J33
DOI: https://doi.org/10.1090/jams/890
Published electronically: November 16, 2017
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Abstract: In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65-168) 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 tropicalizations 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 (Publ. Inst. Hautes Études Sci., 103 (2006), 1-211). 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 $ H^0(Y,\mathcal {O}_Y)$ extending to a basis of $ H^0(U,\mathcal {O}_U)$. 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 $ \operatorname {SL}_r$, parameterized by a set which each choice of seed identifies with the 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.


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Additional Information

Mark Gross
Affiliation: DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Email: mgross@dpmms.cam.ac.uk

Paul Hacking
Affiliation: Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, Massachusetts 01003-9305
Email: hacking@math.umass.edu

Sean Keel
Affiliation: Department of Mathematics, 1 University Station C1200, Austin, Texas 78712-0257
Email: keel@math.utexas.edu

Maxim Kontsevich
Affiliation: IHÉS, Le Bois-Marie 35, route de Chartres, 91440 Bures-sur-Yvette, France
Email: maxim@ihes.fr

DOI: https://doi.org/10.1090/jams/890
Received by editor(s): November 7, 2014
Received by editor(s) in revised form: October 28, 2016, and September 4, 2017
Published electronically: November 16, 2017
Additional Notes: The first author was partially supported by NSF grant DMS-1262531 and a Royal Society Wolfson Research Merit Award, the second by NSF grants DMS-1201439 and DMS-1601065, and the third by NSF grant DMS-0854747. Some of the research was conducted when the first and third authors visited the fourth at I.H.E.S. during the summers of 2012 and 2013.
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