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SIGMA, 2015, Volume 11, 033, 32 pp. (Mi sigma1014)  

This article is cited in 5 scientific papers (total in 5 papers)

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

Yuki Kanakubo, Toshiki Nakashima

Division of Mathematics, Sophia University, Yonban-cho 4, Chiyoda-ku, Tokyo 102-0081, Japan

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 $\bar{\mathcal{A}}(\mathbf{i})_{{\mathbb C}}$ and the generalized minors $\{\Delta(k;{\mathbf{i}})\}$ 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 $G=SL_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe the generalized minors $\{\Delta(k;{\mathbf{i}})\}$ as summations of monomial realizations of certain Demazure crystals.

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

DOI: https://doi.org/10.3842/SIGMA.2015.033

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ArXiv: 1409.8622
MSC: 13F60; 81R50; 17B37
Received: October 1, 2014; in final form April 14, 2015; Published online April 23, 2015
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Citation: Yuki Kanakubo, Toshiki Nakashima, “Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A”, SIGMA, 11 (2015), 033, 32 pp.

Citation in format AMSBIB
\Bibitem{KanNak15}
\by Yuki~Kanakubo, Toshiki~Nakashima
\paper Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type~A
\jour SIGMA
\yr 2015
\vol 11
\papernumber 033
\totalpages 32
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\crossref{https://doi.org/10.3842/SIGMA.2015.033}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3338679}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84929441354}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. Kanakubo, T. Nakashima, “Explicit forms of cluster variables on double Bruhat cells $G^{u,e}$ of type C”, Tokyo J. Math., 39:3 (2017), 643–678  crossref  mathscinet  zmath  isi
    2. Yu. Kanakubo, “Explicit forms of cluster variables on double Bruhat cells $G^{u,e}$ of type B”, Commun. Algebr., 45:9 (2017), 3825–3860  crossref  mathscinet  zmath  isi  scopus
    3. Sh. Zelikson, “Auslander–Reiten numbers games”, Algebr. Represent. Theory, 21:2 (2018), 277–307  crossref  mathscinet  zmath  isi
    4. Yu. Kanakubo, T. Nakashima, “Cluster variables on double Bruhat cells $G^{u,e}$ of classical groups and monomial realizations of Demazure crystals”, Int. Math. Res. Notices, 2018, no. 12, 3621–3670  crossref  isi  scopus
    5. Kanakubo Yu., “Cluster Algebras of Finite Type Via Coxeter Elements and Demazure Crystals of Type B,C,D”, J. Geom. Phys., 137 (2019), 40–75  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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