X=M for symmetric powers

Abstract

The X=M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac--Moody algebra. In this paper we prove the X=M conjecture for tensor products of Kirillov--Reshetikhin crystals B^{1,s} associated to symmetric powers for all nonexceptional affine algebras.

40 pages; to appear in J. Algebra

References in corpus (2)

• Quantum Affine Algebras and their Representations
• Virtual crystals and Kleber's algorithm

Cited by in corpus (26)

• Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection
• A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X
• Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas
• Tau functions in combinatorial Bethe ansatz
• New fermionic formula for unrestricted Kostka polynomials
• Proof of the combinatorial Kirillov-Reshetikhin conjecture
• Affine crystal structure on rigged configurations of type D_n^(1)
• Finite-Dimensional Crystals B^{2,s} for Quantum Affine Algebras of type D_{n}^{(1)}
• Rigged Configurations and Kashiwara Operators
• Crystal Interpretation of Kerov-Kirillov-Reshetikhin Bijection II. Proof for sl_n Case
• Crystal structure on rigged configurations
• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
• Type $D_n^{(1)}$ rigged configuration bijection
• X=M Theorem: Fermionic formulas and rigged configurations under review
• Rigged configurations and the $\ast$-involution
• KKR type bijection for the exceptional affine algebra E_6^{(1)}
• Stable Rigged Configurations for Quantum Affine Algebras of Nonexceptional Types
• Connecting marginally large tableaux and rigged configurations via crystals
• A uniform approach to soliton cellular automata using rigged configurations
• Virtualization map for the Littelmann path model
• Uniform description of the rigged configuration bijection
• Applications of Crystal Bases to Current Problems in Representation Theory
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems
• An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_2^{(1)}$