Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

Abstract

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of Shimozono and Zabrocki (J Algebra 299:33-61, 2006) by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any onedimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.

Demazure crystals and tensor products of perfect Kirillov–Reshetikhin crystals with various levels

“…Proof. Although the proof is carried out in a similar way as that of [21,Lemma 8.2], we give it for the reader's convenience. The case τ = id is trivial.…”

Stable rigged configurations for quantum affine algebras of nonexceptional types

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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.

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Combinatorial extension of stable branching rules for classical groups

“…By using Theorem 5.3, we obtain a new combinatorial formula for (5.4) or a combinatorial extension of (5.5) to arbitrary λ (i) 's. We should remark that a q-analogue of (5.4) is introduced in [28], and its stable limit is closely related with the energy function on a tensor product of finite affine crystals (see for example [30]). It would be interesting to find a combinatorial interpretation of the q-analogue of (5.4) or (5.5) in terms of spinor model.…”