Uniform description of the rigged configuration bijection

• Travis Scrimshaw ^[1]
  1. [1] University of Queensland

     University of Queensland

     Australia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 3, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-00564-8
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• Resumen

  • We give a uniform description of the bijection Φ from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form ⨂Ni=1Bri,1 in dual untwisted types: simply-laced types and types A(2)2n−1, D(2)n+1, E(2)6, and D(3)4. We give a uniform proof that Φ is a bijection and preserves statistics. We describe Φ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that Φ is a bijection for ⨂Ni=1Bri,si when ri, for all i, map to 0 under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov–Reshetikhin crystals Br,1 using tableaux of a fixed height kr depending on r in all affine types. Additionally, we are able to describe crystals Br,s using kr×s shaped tableaux that are conjecturally the crystal basis for Kirillov–Reshetikhin modules for various nodes r.