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Theta bases are atomic

Part of: Arithmetic rings and other special rings

Published online by Cambridge University Press:  26 April 2017

Travis Mandel
Travis Mandel*
Affiliation:
University of Utah, Department of Mathematics, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA email [email protected]
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Abstract

Fock and Goncharov conjectured that the indecomposable universally positive (i.e. atomic) elements of a cluster algebra should form a basis for the algebra. This was shown to be false by Lee, Li and Zelevinsky. However, we find that the theta bases of Gross, Hacking, Keel and Kontsevich do satisfy this conjecture for a slightly modified definition of universal positivity in which one replaces the positive atlas consisting of the clusters by an enlargement we call the scattering atlas. In particular, this uniquely characterizes the theta functions.

Keywords

cluster algebratheta functionscattering diagrampositivitybasis

MSC classification

Primary: 13F60: Cluster algebras
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 6 , June 2017 , pp. 1217 - 1219
Copyright
© The Author 2017 

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References

Cheung, M. W., Gross, M., Muller, G., Musiker, G., Rupel, D., Stella, S. and Williams, H., The greedy basis equals the theta basis: a rank two haiku , J. Combin. Theory Ser. A 145 (2017), 150–171; MR 3551649.CrossRefGoogle Scholar
Davison, B., Positivity for quantum cluster algebras, Preprint (2016), arXiv:1601.07918.Google Scholar
Fock, V. and Goncharov, A., Cluster ensembles, quantization and the dilogarithm , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865–930.Google Scholar
Fock, V. V. and Goncharov, A. B., Cluster Poisson varieties at infinity , Selecta Math. (N.S.) 22 (2016), 2569–2589.CrossRefGoogle Scholar
Gross, M., Hacking, P., Keel, S. and Kontsevich, M., Canonical bases for cluster algebras, Preprint (2014), arXiv:1411.1394.Google Scholar
Lee, K., Li, L. and Zelevinsky, A., Positivity and tameness in rank 2 cluster algebras , J. Algebraic Combin. 40 (2014), 823–840.Google Scholar
Mandel, T., Refined tropical curve counts and canonical bases for quantum cluster algebras, Preprint (2015), arXiv:1503.06183.Google Scholar