A bijection between type crystals and rigged configurations

Hatayama et al. conjectured fermionic formulas associated with tensor products of -crystals Br,s. The crystals Br,s correspond to the Kirillov–Reshetikhin modules which are certain finite-dimensional -modules. In this paper we present a combinatorial description of the affine crystals Br,1 of type . A statistic preserving bijection between crystal paths for these crystals and rigged configurations is given, thereby proving the fermionic formula in this case. This bijection reflects two different methods to solve lattice models in statistical mechanics: the corner-transfer-matrix method and the Bethe Ansatz.