B型双Bruhat胞腔G^{u，e}上簇变量的显式形式
Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type B
摘要：
设$G$是$B_r$型的$\mathbb{C}$上的单连通单代数群，$B$和$B_-$是它的两个相反的Borel子群，$W$是相伴的Weyl群.对于$u$，$v\in W$，已知双Bruhat单元$G^{u,v}=BuB\cap B_-vB_-$的坐标环${\mathbb C}[G^{u,v}]$同构于上簇代数$\overline{\mathcal{A}}(\textbf{i})_{\mathbb C}$，并且广义子式$Δ(k;\textbf{i})$是${\mathbb C}[G^{u,v}]$的簇变量[A.Berenstein，S.Fomin，A.Zelevinsky，Duke Math.J.126（2005），1-52，arxiv：math.RT/0305434]。最近的研究表明，${\mathbb C}[G^{u,v}]$具有簇代数结构[K. R. Goodearl，M. T. Yakimov，arxiv：1602.00498（2016）]。在$v=e$的情况下，我们将明确地描述广义子式$Δ(k;\textbf{i})$。
英文摘要：
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ 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 $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\textbf{i})_{\mathbb C}$ and the generalized minors $Δ(k;\textbf{i})$ are the cluster variables of ${\mathbb C}[G^{u,v}]$[A.Berenstein, S.Fomin, A.Zelevinsky, Duke Math. J. 126 (2005), 1-52, arxiv:math.RT/0305434]. Recently, it is also shown that ${\mathbb C}[G^{u,v}]$ have structure of cluster algebra [K. R. Goodearl, M. T. Yakimov, arxiv:1602.00498 (2016)]. In the case $v=e$, we shall describe the generalized minor $Δ(k;\textbf{i})$ explicitly.
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