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Affine structures and a tableau model for $E_6$ crystals

Abstract : We provide the unique affine crystal structure for type $E_6^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the multiples of fundamental weights $s\Lambda _1, s\Lambda _2$, and $s\Lambda _6$ for all $s≥ 1$ (in Bourbaki's labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type $E$ and use the order three automorphism of the affine $E_6^{(1)}$ Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type $E_7^{(1)}$ Kirillov―Reshetikhin crystals corresponding to the adjoint node.
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Brant Jones, Anne Schilling. Affine structures and a tableau model for $E_6$ crystals. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.785-796. ⟨hal-01186235⟩

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