Ultra-discretization of D(1) 6-geometric crystal at the spin node

Abstract

Let g be an affine Lie algebra with index set I = {0, 1, 2, ···,n}. It is conjectured in [12] that for each Dynkin node k ∈ I\{0} theaffineLiealgebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for the Langland dual gL.In this paper we show that at the spin node k = 6, the family of perfect crystals given in [6] form a coherent family and show that its limit B6,∞ is isomorphic to the ultra-discretization of the positive geometric crystal we constructed in [18] for the affine Lie algebra D(1) 6 which proves the conjecture in this case. © 2021 American Mathematical Society.