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

## TL;DR

This paper characterizes all cluster variables in the coordinate ring of certain double Bruhat cells for classical groups of types B, C, D as monomial realizations of Demazure crystals, linking cluster algebras to crystal theory.

## Contribution

It explicitly describes cluster variables in types B, C, D as monomial realizations of Demazure crystals, extending understanding of cluster structures in classical groups.

## Key findings

- All cluster variables are described as monomial realizations of Demazure crystals.
- The work applies to classical groups of types B, C, D.
- Provides a detailed combinatorial model for cluster variables.

## 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 $G^{e,c^2}:=B\cap B_-c^2B_-$ 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 ${\rm B}_r$, ${\rm C}_r$ or ${\rm D}_r$ and describe all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ as monomial realizations of certain Demazure crystals.