A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X

Abstract

We establish the equality of the specialization $P_{\lambda }(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_{\lambda }(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the $t=0$ limit of the symmetric Macdonald polynomial in the untwisted and non-simply-laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.

43 pages, 1 figure; slight reorganization of the material, additional details added

References in corpus (7)

• BGG reciprocity for current algebras
• X=M for symmetric powers
• Specializations of nonsymmetric Macdonald-Koornwinder polynomials
• Modules with Demazure Flags and Character Formulae
• Highest weight categories and Macdonald polynomials
• Quantum Lakshmibai-Seshadri paths and root operators
• Equivariant K-Chevalley Rules for Kac-Moody Flag Manifolds

Cited by in corpus (21)

• BGG reciprocity for current algebras
• A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at $t=0$ and Demazure characters
• Equivariant $K$-theory of semi-infinite flag manifolds and Pieri-Chevalley formula
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
• Quantum Lakshmibai-Seshadri paths and root operators
• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials
• Generalized Weyl modules and Demazure submodules of level-zero extremal weight modules
• Existence of Kirillov--Reshetikhin crystals for multiplicity free nodes
• Virtualization map for the Littelmann path model
• Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras
• Uniform description of the rigged configuration bijection
• A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
• A Weyl Module Stratification of Integrable Representations (with an appendix by Ryosuke Kodera)
• Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths
• Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths
• Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t=\infty$
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• Quantum Lakshmibai-Seshadri paths and the specialization of Macdonald polynomials at $t=0$ in type $A_{2n}^{(2)}$
• Kirillov-Reshetikhin crystals $B^{1,s}$ for ${\widehat{\mathfrak{sl}}}_n$ using Nakajima monomials