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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References

  1. Berenstein A., Fomin S., Zelevinsky A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52, math.RT/0305434.
  2. Berenstein A., Zelevinsky A., Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128, math.RT/9912012.
  3. Fomin S., Zelevinsky A., Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335-380, math.RT/9802056.
  4. Gekhtman M., Shapiro M., Vainshtein A., Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, Vol. 167, Amer. Math. Soc., Providence, RI, 2010.
  5. Hong J., Kang S.-J., Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, Vol. 42, Amer. Math. Soc., Providence, RI, 2002.
  6. Kashiwara M., On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516.
  7. Kashiwara M., Bases cristallines des groupes quantiques, Cours Spécialisés, Vol. 9, Société Mathématique de France, Paris, 2002.
  8. Kashiwara M., Realizations of crystals, in Combinatorial and Geometric Representation Theory (Seoul, 2001), Contemp. Math., Vol. 325, Amer. Math. Soc., Providence, RI, 2003, 133-139, math.QA/0202268.
  9. Kashiwara M., Nakashima T., Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), 295-345.
  10. Nakajima H., $t$-analogs of $q$-characters of quantum affine algebras of type $A_n$, $D_n$, in Combinatorial and Geometric Representation Theory (Seoul, 2001), Contemp. Math., Vol. 325, Amer. Math. Soc., Providence, RI, 2003, 141-160, math.QA/0204184.
  11. Nakashima T., Decorations on geometric crystals and monomial realizations of crystal bases for classical groups, J. Algebra 399 (2014), 712-769, arXiv:1301.7301.

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