Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups

Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety,...

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Bibliographic Details
Published in:arXiv.org 2016-09
Main Author: Weng, Daping
Format: Article
Language:English
Subjects:
Automorphisms
Clusters
Coding
Invariants
Toruses
Transformations
Online Access:Get full text
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Summary:Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as the Donaldson-Thomas transformation. In this paper we prove a conjecture of Goncharov and Shen in the case of \(\mathrm{GL}_n\), which describes the Donaldson-Thomas transformation of the double quotient of the double Bruhat cells \(H \backslash \mathrm{GL}_n^{u,v}/H\) where \(H\) is a maximal torus, as a certain explicit cluster transformation related to Fomin-Zelevinsky's twist map. Our result, combined with the work of Gross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of \(H\backslash \mathrm{GL}_n^{u,v}/H\).
ISSN:2331-8422