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 Br,1Br,1 and B1,sB1,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 Br,sBr,s, show that it preserves statistics, and obtain the so-called Kirillov–Reshetikhin tableaux model for Br,sBr,s. Additionally, we show that Φ commutes with the virtualization map and that B1,sB1,s is naturally a virtual crystal in type D4(1), thus defining an affine crystal structure on rigged configurations corresponding to B1,sB1,s.