2024/04/09 by Dinkins, Erica S., Misra, Kailash C.
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2404.06321
Let \mathfrakg be an affine Lie algebra with index set I = \0, 1, 2, ⋯ , n\ and \mathfrakgL be its Langlands dual. It is conjectured that for each Dynkin node i ∈ I ∖ \0\ the affine Lie algebra \mathfrakg has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual \mathfrakgL. In this paper we construct positive geometric crystals for V(Cn(1)) in the level zero fundamental spin Cn(1)- module W(\varpin) for n = 2, 3,4 and show that its ultra-discretization is isomorphic to the limit Bn, ∞ of a coherent family \Bn, l\l ≥ 1 of perfect crystals for the Langland dual Dn(2) which proves the conjecture in these cases.