A crystal to rigged configuration bijection and the filling map for type D-4(3)

We give a bijection Phi from rigged configurations to a tensor product of Kirillov Reshetikhin crystals of the form B-r,B-1 and B-1,B-s in type D-4((3)). We show that the cocharge statistic is sent to the energy statistic for tensor products circle times(N)(i=1) B-r,B-1 and circle times(N)(i=1) B-1,B-s. We extend this bijection to a single B-r,B-s, show that it preserves statistics, and obtain the so-called Kirillov-Reshetikhin tableaux model for B-r,B-s. Additionally, we show that Phi commutes with the virtualization map and that B-1,B-s is naturally a virtual crystal in type D-4((1)), thus defining an affine crystal structure on rigged configurations corresponding to B-1,B-s. (C) 2015 Elsevier Inc. All rights reserved