A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$

Abstract

We give a bijection $\Phi$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products ${⨂}_{i=1}^N{B}^{r_i,1}$ and ${⨂}_{i=1}^N{B}^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show that $\Phi$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in type $D_4^{(1)}$, thus defining an affine crystal structure on rigged configurations corresponding to $B^{1,s}$.

40 pages, 6 figures; various revisions from referee comments and fixed minor typos

References in corpus (6)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• X=M for symmetric powers
• Perfect Crystals for U_q(D_4^{(3)})
• Kirillov--Reshetikhin modules associated to G_2
• A rigged configuration model for $B(\infty )$
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Cited by in corpus (8)

• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• Type $D_n^{(1)}$ rigged configuration bijection
• Rigged configurations and the $\ast$-involution
• Connecting marginally large tableaux and rigged configurations via crystals
• A uniform approach to soliton cellular automata using rigged configurations
• Virtualization map for the Littelmann path model
• Uniform description of the rigged configuration bijection
• An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_2^{(1)}$