Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type B,C,D

Abstract

For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_{-}{c}^2{B}_{-}$ has a structure of cluster algebra of finite type, where $B$ and $B_{-}$ are opposite Borel subgroups. In this article, we consider the case $G$ is of type ${\mathrm{B}}_r$, ${\mathrm{C}}_r$ or ${\mathrm{D}}_r$ and describe all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ as monomial realizations of certain Demazure crystals.

arXiv admin note: text overlap with arXiv:1703.08323, arXiv:1604.05956

References in corpus (3)

• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A
• Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Classical Groups and Monomial Realizations of Demazure Crystals