Resumen de Uniform description of the rigged configuration bijection

Travis Scrimshaw

• 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.