Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 033, 32 pages      arXiv:1409.8622      https://doi.org/10.3842/SIGMA.2015.033
Contribution to the Special Issue on New Directions in Lie Theory

Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A

Yuki Kanakubo and Toshiki Nakashima
Division of Mathematics, Sophia University, Yonban-cho 4, Chiyoda-ku, Tokyo 102-0081, Japan

Received October 01, 2014, in final form April 14, 2015; Published online April 23, 2015

Abstract
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the 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 $\bar{{\mathcal A}}({\bf i})_{{\mathbb C}}$ and the generalized minors $\{\Delta(k;{\bf i})\}$ are the cluster variables belonging to a given initial seed in ${\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case $G={\rm SL}_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe the generalized minors $\{\Delta(k;{\bf i})\}$ as summations of monomial realizations of certain Demazure crystals.

Key words: cluster variables; double Bruhat cells; crystal bases; monomial realizations, generalized minors.

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