Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection

Abstract

The Kerov-Kirillov-Reshetikhin (KKR) bijection is the crux in proving fermionic formulas. It is defined by a combinatorial algorithm on rigged configurations and highest paths. We reformulate the KKR bijection as a vertex operator by purely using combinatorial R in crystal base theory. The result is viewed as a nested Bethe ansatz at q=0 as well as the direct and the inverse scattering (Gel'fand-Levitan) map in the associated soliton cellular automaton.

arXiv comment28 pages

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• Kirillov--Schilling--Shimozono bijection as energy functions of crystals
• Dynamics of the box-ball system with random initial conditions via Pitman's transformation
• Crystal Interpretation of Kerov-Kirillov-Reshetikhin Bijection II. Proof for sl_n Case
• Rigged Configurations and Kashiwara Operators
• Randomized box-ball systems, limit shape of rigged configurations and Thermodynamic Bethe ansatz
• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve
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• Inhomogeneous generalization of multispecies totally asymmetric zero range process
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• Box-Basket-Ball Systems
• Stable Rigged Configurations for Quantum Affine Algebras of Nonexceptional Types
• Combinatorial Bethe ansatz and generalized periodic box-ball system
• A uniform approach to soliton cellular automata using rigged configurations
• Ultradiscrete Soliton Systems and Combinatorial Representation Theory
• Finding Rigged Configurations From Paths
• Rigged configurations as tropicalizations of loop Schur functions
• Uniform description of the rigged configuration bijection
• Generalized hydrodynamics in complete box-ball system for $U_q({\widehat{sl}}_n)$
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• Linearization of the box-ball system: an elementary approach
• Scaling limit of soliton lengths in a multicolor box-ball system
• A Path-Counting Analysis of Phase Shifts in Box-Ball Systems
• A crystal theoretic method for finding rigged configurations from paths
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• Similarity and Kirillov-Schilling-Shimozono bijection