Stable rigged configurations for quantum affine algebras of nonexceptional types

Abstract

For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood-Richardson coefficients. Combining this result with [13] and [19] we settle the X = M conjecture under the large rank hypothesis.

Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

“…n in [15]. It should be mentioned that this conjecture was recently settled for nonexceptional g under the large rank hypothesis by combining the results in [15], [18] and [22], and for general g if ℓ 1 = • • • = ℓ p = 1 [20].…”

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

“…We remark that these results can be viewed as another important example of significant properties of rigged configurations with respect to deep structures of the underlying algebra. For example, in [19] an interesting new bijection related to rigged configurations and Littlewood-Richardson tableaux is introduced which is expected to be an analogue of the involution corresponding to exchanging Dynkin nodes 0 and n − 1 constructed in [15]. Another such phenomenon is that generalizations of Schützenberger's involution become simple operations on rigged configurations (taking complements of the riggings, see [28]).…”

Rigged Configurations and Kashiwara Operators

“…Remarkably, the construction is uniform for all types of algebras and the Littlewood-Richardson tableaux naturally appear as the recording tableaux of the algorithm. We expect that the bijection coincides with (and generalizes) a canonical Dynkin diagram automorphism (see Remark 3.2 of [23]).…”

“…Step 1. Suppose that we have P (i) 23, we see that f i acts on the same string before and after δ. Let us consider the behavior of δ before and after f i .…”