Combinatorial structure of Kirillov–Reshetikhin crystals of type , ,

We provide the explicit combinatorial structure of the Kirillov–Reshetikhin crystals Br,s of type , , and . This is achieved by constructing the crystal analogue σ of the automorphism of the (respectively or ) Dynkin diagram that interchanges the 0 and 1 node. The involution σ is defined in terms of new ± diagrams that govern the Dn to Dn−1 (respectively Bn to Bn−1, or Cn to Cn−1) branching. It is also shown that the crystal Br,s is perfect. These crystals have been implemented in MuPAD-Combinat; the implementation is discussed in terms of many examples.