Publications (28)34.76 Total impact
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ABSTRACT: We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).  [Show abstract] [Hide abstract]
ABSTRACT: We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin1/2 isotropic Heisenberg model. We analyze the non selfconjugate solutions of DeguchiGiri and the counter example for the string hypothesis discovered by EsslerKorepinSchoutens. In both cases clear discrete structures appear.  [Show abstract] [Hide abstract]
ABSTRACT: Rigged configurations are known to provide actionangle variables for remarkable discrete dynamical systems known as boxball systems. We conjecture an explicit piecewiselinear formula to obtain the shapes of a rigged configuration from a tensor product of onerow crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric Rmatrix. Our piecewiselinear formula is obtained as the tropicalization of ratios of cylindric loop Schur functions. We prove our conjecture for the first shape of a rigged configuration, thus giving a piecewiselinear formula for the lengths of the solitons of a boxball system.  [Show abstract] [Hide abstract]
ABSTRACT: We compute the energy eigenvalues of NepomechieWang's eigenstates for the spin 1/2 isotropic Heisenberg chain.  [Show abstract] [Hide abstract]
ABSTRACT: We provide a conjecture for the following two quantities related with the spin$\frac{1}{2}$ isotropic Heisenberg model defined over rings of even lengths: (i) the number of the solutions to the Bethe ansatz equations which correspond to nonzero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of NepomechieWang. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.  [Show abstract] [Hide abstract]
ABSTRACT: For types A^{(1)}_n and D^{(1)}_n we prove that the rigged configuration bijection intertwines the classical Kashiwara operators on tensor products of the arbitrary KirillovReshetikhin crystals and the set of the rigged configurations. 
Article: BoxBasket Systems
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ABSTRACT: Using the whurl relation of the first two authors, we define a new discrete solitonic system, which we call the boxbasketball system, generalizing the boxball system of Takahashi and Satsuma. In boxbasketball systems, balls may be put either into boxes or into baskets. While boxes stay fixed, both balls and baskets get moved during time evolution. Balls and baskets behave as fermionic and bosonic particles, respectively. We classify the solitons of this system, and study their scattering.  [Show abstract] [Hide abstract]
ABSTRACT: For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood–Richardson coefficients. Combining this result with theorems of Kirillov–Schilling–Shimozono and Lecouvey–Okado–Shimozono, we settle the X = M conjecture under the large rank hypothesis. Résumé. Pour uneal ebre affine de type nonexceptionnel de grand rang nous prouvons que la formule fermionique dépend seulement du voisinage du noeud 0 dans le diagramme de Dynkin, e egalement que la formule fermionique en type autre que A peu etre exprimée comme combinaison de celles de type A avec des coefficients de Littlewood– Richardson. Combinant ce résultat avec desthéo emes de Kirillov–Schilling–Shimozono et de Lecouvey–Okado– Shimozono, nous résolvons la conjecture X = M lorsque le rang est grand.  [Show abstract] [Hide abstract]
ABSTRACT: We introduce generalized energies for a class of Uq(D(1)n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and antiparticles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable D(1)n cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions.  [Show abstract] [Hide abstract]
ABSTRACT: Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the KirillovReshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of B^{r,s} (r not equal to n1,n) in terms of tableaux of rectangular shape r x s, which we coin KirillovReshetikhin tableaux (using a nontrivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of KirillovReshetikhin crystals and rigged configurations. 
Article: BoxBasketBall Systems
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ABSTRACT: For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with LittlewoodRichardson coefficients. Combining this result with math.CO/9901037 and arXiv:1002.3715 we settle the X=M conjecture under the large rank hypothesis.  [Show abstract] [Hide abstract]
ABSTRACT: We introduce generalized energies for a class of U_q(D^{(1)}_n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and antiparticles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable D^{(1)}_n cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions. Comment: For proceedings of ``Infinite Analysis 09: New Trends in Quantum Integrable Systems"  [Show abstract] [Hide abstract]
ABSTRACT: We give an interpretation of the t=1 specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of BoxBall Systems (BBSpaths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBSpaths which all, conjecturally, have the same distribution. Comment: 15 pages, typos corrected, French abstract added, version for publication in proceedings of FPSAC 2010.  [Show abstract] [Hide abstract]
ABSTRACT: We calculate the image of the combinatorial Rmatrix for any classical highest weight element in the tensor product of KirillovReshetikhin crystals $B^{r,k}\otimes B^{1,l}$ of type $D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n1}$. The notion of $\pm$diagrams is effectively used for the identification of classical highest weight elements in $B^{1,l}\otimes B^{r,k}$. Comment: 27 pages  [Show abstract] [Hide abstract]
ABSTRACT: There are two distinct approaches to the study of initial value problem of the periodic boxball systems. One way is the rigged configuration approach due to KunibaTakagiTakenouchi and another way is the 10elimination approach due to MadaIdzumiTokihiro. In this paper, we describe precisely interrelations between these two approaches. Comment: 16 pages, final version, minor revision  [Show abstract] [Hide abstract]
ABSTRACT: We give several equivalent combinatorial descriptions of the space of states for the boxball systems, and connect certain partition functions for these models with the qweight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t=1 of the HaglundHaimanLoehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the socalled energy statistics. Comment: 35 pages, minor revision, final version  [Show abstract] [Hide abstract]
ABSTRACT: We reformulate the KerovKirillovReshetikhin (KKR) map in the combinatorial Bethe ansatz from paths to rigged configurations by introducing local energy distribution in crystal base theory. Combined with an earlier result on the inverse map, it completes the crystal interpretation of the KKR bijection for Uq(widehat { {sl}}2). As an application, we solve an integrable cellular automaton, a higher spin generalization of the periodic boxball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function.  [Show abstract] [Hide abstract]
ABSTRACT: We review reformulation of the map from tensor product of crystals to the rigged configurations in terms of the energy function of affine crystals. Especially, we give intuitive picture of the inverse scattering formalism for the periodic boxball systems formulated by KunibaTakagiTakenouchi (arXiv:math/0602481v2).  [Show abstract] [Hide abstract]
ABSTRACT: In proving the Fermionic formulae, a 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 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.
Publication Stats
182  Citations  
34.76  Total Impact Points  
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Institutions

20102015

Tokyo University of Science
Edo, Tōkyō, Japan


20042009

The University of Tokyo
 Department of Physics
Edo, Tōkyō, Japan
