Publications (26)32.55 Total impact
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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.10/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We compute the energy eigenvalues of NepomechieWang's eigenstates for the spin 1/2 isotropic Heisenberg chain.06/2014;  [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.Journal of Physics A Mathematical and Theoretical 02/2014; 47(20). · 1.77 Impact Factor  [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.Symmetry Integrability and Geometry Methods and Applications 02/2013; · 1.30 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This lecture note is intended to be a brief introduction to a recent development on the interplay between the ultradiscrete (or tropical) soliton systems and the combinatorial representation theory. We will concentrate on the simplest cases which admit elementary explanations without losing essential ideas of the theory. In particular we give definitions for the main constructions corresponding to the vector representation of type $A^{(1)}_1$.12/2012; 
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.Reviews in Mathematical Physics 09/2012; 24(8):50019. · 1.45 Impact Factor  [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.10/2011;  [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.Journal of Algebraic Combinatorics 09/2011; · 0.72 Impact Factor  [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 LittlewoodRichardson coefficients. Combining this result with math.CO/9901037 and arXiv:1002.3715 we settle the X=M conjecture under the large rank hypothesis.Advances in Mathematics 08/2010; · 1.35 Impact Factor  [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"01/2010;  [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 2010.12/2009;  [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 pagesInternational Mathematics Research Notices 03/2009; · 1.07 Impact Factor  [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 revisionLetters in Mathematical Physics 02/2009; · 2.07 Impact Factor  [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 versionMoscow Mathematical Journal 11/2008; · 0.35 Impact Factor  [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.Reviews in Mathematical Physics 06/2008; 20(05):493527. · 1.45 Impact Factor  [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).05/2008;  [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.Journal of Algebraic Combinatorics 01/2008; 27(1):5598. · 0.72 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The KirillovSchillingShimozono (KSS) bijection appearing in theory of the Fermionic formula gives one to one correspondence between the set of elements of tensor products of the KirillovReshetikhin crystals (called paths) and the set of rigged configurations. It is generalization of KerovKirillovReshetikhin bijection and plays inverse scattering formalism for the boxball systems. In this paper, we give algebraic reformulation of the KSS map from the paths to rigged configurations, using the combinatorial R and energy functions of crystals. It gives characterization of the KSS bijection as intrinsic property of tensor products of crystals.International Mathematics Research Notices 12/2007; · 1.07 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We introduce ultradiscrete tau functions associated with rigged configurations for . 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 Nsoliton solutions of the box–ball system.Nuclear Physics B 12/2007; · 3.95 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The KerovKirillovReshetikhin (KKR) bijection gives one to one correspondences between the set of highest paths and the set of rigged configurations. In this paper, we give a crystal theoretic reformulation of the KKR map from the paths to rigged configurations, using the combinatorial R and energy functions. This formalism provides tool for analysis of the periodic boxball systems.Journal of Physics A Mathematical and Theoretical 09/2007; · 1.77 Impact Factor
Publication Stats
145  Citations  
32.55  Total Impact Points  
Top Journals
Institutions

2010–2014

Tokyo University of Science
Edo, Tōkyō, Japan


2004–2009

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


2007

Kobe University
 Department of Mathematics
Kōbe, Hyōgo, Japan
