2016/08/22 by Kailash C. Misra, Toshiki Nakashima, Misra, Kailash C. +1
Mathematics · #14M14 #17B37 #17B67 #22E65 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometry and complex manifolds #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:14M14 #msc:17B37 #msc:17B67 #msc:22E65
paper · pdf · doi:10.48550/arxiv.1608.06063
42 pages. arXiv admin note: text overlap with arXiv:1209.4565
arxiv created 2016/08/22 · openalex publication_date 2016/08/22 · arxiv updated 2016/08/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let \mathfrak g be an affine Lie algebra with index set I = \0, 1, 2, ⋯ , n\ and \mathfrak gL be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each k ∈ I ∖ \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 gL. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra \mathfrak g= A(1)n for each Dynkin index k∈ I∖\0\ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for A(1)n given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.