Affine Geometric Crystal of $A_n^{(1)}$ and Limit of Kirillov-Reshetikhin Perfect Crystals

Abstract

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I=\{0,1,2,\cdots ,n\}$ and ${\mathfrak{g}}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k\in I\setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak{g}}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra $\mathfrak{g}=A_n^{(1)}$ for each Dynkin index $k\in I\setminus \{0\}$ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for $A_n^{(1)}$ given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.

42 pages. arXiv admin note: text overlap with arXiv:1209.4565

References in corpus (1)

• Kirillov--Reshetikhin crystals for nonexceptional types