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

Abstract

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 $H\backslash {\mathrm{GL}}_n^{u,v}/H$.

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Total positivity, Grassmannians, and networks
• Quiver mutation and combinatorial DT-invariants
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• Donaldson-Thomas Transformation of Grassmannian
• Cluster algebras III: Upper bounds and double Bruhat cells

Cited by in corpus (2)

• A survey on maximal green sequences
• Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups