Geodesic
Affine structures and a tableau model for $E_6$ crystals
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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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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     author = {Jones, Brant and Schilling, Anne},
     title = {Affine structures and a tableau model for $E_6$ crystals},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2809},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2809/}
}
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Jones, Brant; Schilling, Anne. Affine structures and a tableau model for $E_6$ crystals. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2809. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2809/

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