Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection

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.Comment: 28 page

Inverse scattering method for a soliton cellular automaton

A set of action-angle variables for a soliton cellular automaton is obtained. It is identified with the rigged configuration, a well-known object in Bethe ansatz. Regarding it as the set of scattering data an inverse scattering method to solve initial value problems of this automaton is presented. By considering partition functions for this system a new interpretation of a fermionic character formula is obtained.Comment: 29 pages, LaTe

Simple Algorithm for Factorized Dynamics of g_n-Automaton

We present an elementary algorithm for the dynamics of recently introduced soliton cellular automata associated with quantum affine algebra U_q(g_n) at q=0. For g_n = A^{(1)}_n, the rule reproduces the ball-moving algorithm in Takahashi-Satsuma's box-ball system. For non-exceptional g_n other than A^{(1)}_n, it is described as a motion of particles and anti-particles which undergo pair-annihilation and creation through a neutral bound state. The algorithm is formulated without using representation theory nor crystal basis theory.Comment: LaTex2e 9 pages, no figure. For proceedings of SIDE IV conferenc

Soliton Cellular Automata Associated With Crystal Bases

We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for \hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(C^{(1)}_{n-1})-crystals.Comment: 29 pages, 1 figure, LaTeX2

Tau functions in combinatorial Bethe ansatz

We introduce ultradiscrete tau functions associated with rigged configurations for A^{(1)}_n. They satisfy an ultradiscrete version of the Hirota bilinear equation and play a role analogous to a corner transfer matrix for the box-ball system. As an application, we establish a piecewise linear formula for the Kerov-Kirillov-Reshetikhin bijection in the combinatorial Bethe ansatz. They also lead to general N-soliton solutions of the box-ball system.Comment: 52 page

Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of Uq(D4(3))U_q(D_4^{(3)})

In terms of the crystal base of a quantum affine algebra $U_q(\mathfrak{g})$, we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra $\mathfrak{g}=D_4^{(3)}$. The solitons therein are labeled by the crystals of quantum affine algebra $U_q(A_1^{(1)})$. The scatteing rule is identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)})$-crystals. Remarkably, the phase shifts in our SCA are given by {\em 3-times} of those in the well-known box-ball system.Comment: 25 page

X=M for symmetric powers

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.Comment: 40 pages; to appear in J. Algebr

New fermionic formula for unrestricted Kostka polynomials

A new fermionic formula for the unrestricted Kostka polynomials of type $A_{n-1}^{(1)}$ is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetihkin modules, not just for the symmetric and anti-symmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.Comment: 35 pages; reference adde

Crystal Interpretation of Kerov-Kirillov-Reshetikhin Bijection II. Proof for sl_n Case

In proving the Fermionic formulae, combinatorial bijection called the Kerov--Kirillov--Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I (math.QA/0601630) written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial $R$ matrices on rigged configurations.Comment: 45 pages, version for publication. Introduction revised, more explanations added to the main tex