Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)

Details Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2) Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)

2007-04-16

arxiv.org/abs/0704.2046v3

J.Algebra319:2938-2962,2008

Mathematics

Quantum Algebra

23 pages; version 3: references added, role of [27] clarified

Scientific paper

10.1016/j.jalgebra.2007.10.020

We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp. B_n(1) or A_{2n-1}(2)) Dynkin diagram that interchanges the 0 and 1 node. The involution sigma is defined in terms of new plus-minus diagrams that govern the D_n to D_{n-1} (resp. B_n to B_{n-1}, or C_n to C_{n-1}) branching. It is also shown that the crystal B^{r,s} is perfect. These crystals have been implemented in MuPAD-Combinat; the implementation is discussed in terms of many examples.

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