Twist automorphisms on quantum unipotent cells and dual canonical bases

Abstract

In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. Moreover, we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.

v3: 57 pages. Major revision. We have detailed our explanation of the classical counterpart of the De Concini-Procesi isomorphisms and the proof of Theorem 7.25. The organization of the paper has been changed. The main results are unchanged v4: 53 pages. Minor corrections. Final version