Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type C

Abstract

Let $G=S{p}_{2r}(\mathbb{C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_{-}$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring $\mathbb{C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_{-}v{B}_{-}$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\text{i}{)}_{\mathbb{C}}$ and the generalized minors $\Delta (k;\text{i})$ are the cluster variables of $\mathbb{C}[G^{u,v}]$[Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $v=e$, we shall describe the generalized minor $\Delta (k;\text{i})$ explicitly.

arXiv admin note: text overlap with arXiv:1409.8622. text overlap with arXiv:1409.8622

References in corpus (2)

• 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

Cited by in corpus (1)

• Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Classical Groups and Monomial Realizations of Demazure Crystals