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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 mandel@math.utah.edu
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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

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 6 , June 2017 , pp. 1217 - 1219
Copyright
© The Author 2017 

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References

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