We give a uniform description of the bijection \(\Phi \) from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form \(\bigotimes _{i=1}^N B^{r_i,1}\) in dual untwisted types: simply-laced types and types \(A_{2n-1}^{(2)}\), \(D_{n+1}^{(2)}\), \(E_6^{(2)}\), and \(D_4^{(3)}\). We give a uniform proof that \(\Phi \) is a bijection and preserves statistics. We describe \(\Phi \) uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that \(\Phi \) is a bijection for \(\bigotimes _{i=1}^N B^{r_i,s_i}\) when \(r_i\), for all i, map to 0 under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov–Reshetikhin crystals \(B^{r,1}\) using tableaux of a fixed height \(k_r\) depending on r in all affine types. Additionally, we are able to describe crystals \(B^{r,s}\) using \(k_r \times s\) shaped tableaux that are conjecturally the crystal basis for Kirillov–Reshetikhin modules for various nodes r.

中文翻译：

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绑定配置双射的统一描述

我们给双射的一个统一的描述\（\披\）从受操纵配置的形式的基里洛夫-Reshetikhin晶体的张量积\（\ bigotimes _ {i = 1} ^ NB ^ {R_I，1} \）在双非扭曲类型：简单花边类型和类型\（A_ {2n-1} ^ {（2）} \），\（D_ {n + 1} ^ {（2）} \），\（E_6 ^ {（2 ）} \）和\（D_4 ^ {（3）} \）。我们给出统一的证明，证明\（\ Phi \）是双射并保留统计信息。我们对所有剩余类型使用虚拟晶体统一描述\（\ Phi \），但是我们的证明是特定于类型的。我们还给出统一的证明，证明\（\ Phi \）是\（\ bigotimes _ {i = 1} ^ NB ^ {r_i，s_i} \）当\（r_i \）对所有i而言，在Dynkin图的自同构下映射为0。此外，我们得到基里洛夫-Reshetikhin晶体的描述\（B ^ {R，1} \）使用一个固定的高度的静态画面\（K_R \）取决于- [R在所有仿射类型。此外，我们能够使用\（k_r \ times s \）形的平台描述晶体\（B ^ {r，s} \），这可以推测是各个节点r的Kirillov–Reshetikhin模块的晶体基础。