A generalization of adjoint crystals for the quantized affine algebras of type A n (1) , C n (1) and D n+1 (2)

Abstract

We generalize Benkart-Frenkel-Kang-Lee's adjoint crystals and describe their crystal structure for type A (1) n , C (1) n and D (2) n+1 .

Uniform description of the rigged configuration bijection

“…However, the U ′ q (g)-crystal structure is not known in general, much less uniformly. One potential approach might be to generalize the approach of [Kod09] by examining various embeddings of B θ,s−1 into B θ,s , where the difficulty is overcoming that the multiplicity of the weights of B(kθ) that do not appear in B (k − 1)θ are not all 1 in general.…”

Quiver Varieties and Adjoint Crystals of Level ℓ for TypeAn(1)

“…In this section, we give a description of adjoint crystals B ad,ℓ for type A (1) n in terms of Young tableaux [18]. Let U q ( • g) be the subalgebra of U q (g) generated by f i , e i and q hi for i ∈ I \ {0}, and let…”

“…It was expected that the crystal B ad = B 0 + B + · · · + B is a perfect crystal of level . This conjecture was proved for types A 1 n C 1 n A 2 2n D 2 n+1 , and D 1 n which yield types B 1 n A 2 2n−1 in [9,11,18,26]. These perfect crystals are called the adjoint crystals of level .…”

“…Then we have the path realization of B arising from B. In this section, we give a description of adjoint crystals B ad for type A 1 n in terms of Young tableaux [18]. Let U q be the subalgebra of U q generated by f i e i and q h i for i ∈ I\ 0 , and let i = i − 0 1 ≤ i ≤ n .…”

Affine structures and a tableau model for E6 crystals

“…A generalization to higher level s for certain nonexceptional types was studied in [Kod08]. These crystals B of level s have the following decomposition when removing the zero arrows [Cha01]:…”