Geodesic
Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection. II: Proof for $\mathfrak{sl}_{n}$ case
Journal of Algebraic Combinatorics, Tome 27 (2008) no. 1, pp. 55-98.

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Summary: In proving the Fermionic formulae, a combinatorial bijection called the Kerov-Kirillov-Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial $R$ matrices on rigged configurations.
Keywords: keywords fermionic formulae, kerov-kirillov-reshetikhin bijection, rigged configuration, crystal bases of quantum affine Lie algebras, box-ball systems, ultradiscrete soliton systems 
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     author = {Sakamoto, Reiho},
     title = {Crystal interpretation of {Kerov-Kirillov-Reshetikhin} bijection. {II:} {Proof} for $\mathfrak{sl}_{n}$ case},
     journal = {Journal of Algebraic Combinatorics},
     pages = {55--98},
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Sakamoto, Reiho. Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection. II: Proof for $\mathfrak{sl}_{n}$ case. Journal of Algebraic Combinatorics, Tome 27 (2008) no. 1, pp. 55-98. http://geodesic.mathdoc.fr/item/JAC_2008__27_1_a2/