Geometric crystals and cluster ensembles in Kac-Moody setting

JOURNAL OF GEOMETRY AND PHYSICS

卷:149

Article

Kanakubo, Yuki¹  Nakashima, Toshiki¹ 

[1] Sophia Univ, Div Math, Chiyoda Ku, Kioicho 7-1, Tokyo 1028554, Japan

关键词: Cluster ensembles;    Double Bruhat cells;    Geometric crystals;    Crystals;   

DOI  :  10.1016/j.geomphys.2019.103576

来源: Elsevier

【 摘 要 】

For a Kac-Moody group G, double Bruhat cells G(u,e) (u is a Weyl group element) have positive geometric crystal structures. In Williams (2013), it is shown that there exist birational maps between 'cluster tori' X-Sigma (resp. A(Sigma)) and G(Ad)(u,e) (resp. G(u,e)), and they are extended to regular maps from cluster X (resp.A)-varieties to G(Ad)(u,e) (resp. G(u,e)). The aim of this article is to compute explicit formulae for the induced geometric crystal structures on these cluster tori. As a corollary, the sets of Z(T)-valued points of these cluster varieties have crystal structures. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free