Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

Abstract

On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A_n^{(1)}$ via polytopes.

References in corpus (6)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Kirillov--Reshetikhin crystals for nonexceptional types
• Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)
• Kirillov--Reshetikhin modules associated to G_2
• RSK correspondence and classically irreducible Kirillov-Reshetikhin crystals
• On the uniqueness of promotion operators on tensor products of type A crystals

Cited by in corpus (4)

• PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams
• FFLV-type monomial bases for type $B$
• Lusztig polytopes and FFLV polytopes
• Kirillov-Reshetikhin crystals $B^{1,s}$ for ${\widehat{\mathfrak{sl}}}_n$ using Nakajima monomials