Affine crystal structure on rigged configurations of type $D_{n}^{(1)}$

Abstract

Extending the work in Schilling (Int. Math. Res. Not. 2006:97376, 2006, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B r,s of type D (1) n for any r, s. We also introduce a representation of B r,s (r = n − 1, n) in terms of tableaux of rectangular shape r × s, which we coin Kirillov-Reshetikhin tableaux (using a nontrivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

“…Left and right-split on B r,s . In [OSS13], we defined the filling map fill on B r,s for 1 ≤ r ≤ n − 2, s ≥ 1. For an I 0 -highest weight element u λ , fill(u λ ) is a tableau of rectangular shape (s r ) which does not necessarily satisfy the conditions of KN tableaux in general.…”

“…We can perform the composition δ • β • γ at once (see, for example, [OSS13,Sak14]). Suppose that we consider B r,s with s > 1.…”

“…In type D (1) n , the bijection for k i=1 B ri,1 was constructed in [Sch05] and for the single KR crystal B = B r,s in [OSS13]. In fact, we rely on these two papers for many properties of combinatorial procedures used in this paper.…”

“…We consider the general case B = B r k ,s k ⊗ · · · ⊗ B r2,s1 ⊗ B r1,s1 . As is revealed in [OSS13], we regard an element of the single KR crystal B r,s as a tableau, called a Kirillov-Reshetikhin tableau, of r × s rectangular shape. For r k ≤ n − 2, we then define three procedures ls, lb, lh on an element of B.…”

Box-Basket-Ball Systems

“…Such a generalization of the rigged configurations have also proven to be useful on representation theory side. Indeed, in [11], the rigged configuration gives an interesting insight into the crystal structure of the KirillovReshetikhin crystals of type D (1) n introduced by [16]. These developments show that the BBS is not only a model in mathematical physics but also gives a source for future mathematical theories.…”