TY - JOUR
T1 - Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases
AU - Kimura, Yoshiyuki
AU - Oya, Hironori
N1 - Funding Information:
This work was supported by JSPS Grant-in-Aid for Scientific Research (S) [24224001 to Y.K.] and JSPS Grant-in-Aid for Young Scientists (B) [17K14168 to Y.K.]; Grant-in-Aid for JSPS Fellows [15J09231 to H.O.]; Program for Leading Graduate Schools, MEXT, Japan [to H.O.]; European Research Council under the European Unions Framework Programme H2020 [647353 Qaffine to H.O.].
Publisher Copyright:
© The Author(s) 2019. Published by Oxford University Press. All rights reserved.
PY - 2021/5/1
Y1 - 2021/5/1
N2 - 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-Schröers description, and Geiß-Leclerc-Schröers 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.
AB - 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-Schröers description, and Geiß-Leclerc-Schröers 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.
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U2 - 10.1093/imrn/rnz040
DO - 10.1093/imrn/rnz040
M3 - Article
AN - SCOPUS:85122189441
SN - 1073-7928
VL - 2021
SP - 6772
EP - 6847
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 9
ER -