A crystal to rigged configuration bijection and the filling map for type D₄⁽³⁾

We give a bijection Φ from rigged configurations to a tensor product of Kirillov-Reshetikhin crystals of the form B^r,1 and B^1,s in type D4(3). We show that the cocharge statistic is sent to the energy statistic for tensor products i=1NBri,1 and i=1NB1,si. 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 Φ commutes with the virtualization map and that B^1,s is naturally a virtual crystal in type D4(1), thus defining an affine crystal structure on rigged configurations corresponding to B^1,s.