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

Abstract

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.

arXiv comment23 pages; version 3: references added, role of [27] clarified

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Cited by in corpus (14)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Kirillov--Reshetikhin crystals for nonexceptional types
• Affine structures and a tableau model for E_6 crystals
• Affine crystal structure on rigged configurations of type D_n^(1)
• Rigged Configurations and Kashiwara Operators
• Realization of affine type A Kirillov-Reshetikhin crystals via polytopes
• Type $D_n^{(1)}$ rigged configuration bijection
• Soliton cellular automaton associated with $G_2^{(1)}$ crystal base
• Box-Basket-Ball Systems
• Combinatorial R-matrices for Kirillov--Reshetikhin crystals of type D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}
• On multigraded generalizations of Kirillov-Reshetikhin modules
• Quantum extremal loop weight modules and monomial crystals
• Soliton cellular automaton associated with $D_n^{(1)}$-crystal $B^{2,s}$
• Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function