Reiho Sakamoto

Tokyo University of Science, Edo, Tōkyō, Japan

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Publications (28)34.76 Total impact

  • Anatol N. Kirillov · Reiho Sakamoto
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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 spin-1/2 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).
    No preview · Article · Sep 2015
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    Reiho Sakamoto
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    ABSTRACT: We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model. We analyze the non self-conjugate solutions of Deguchi-Giri and the counter example for the string hypothesis discovered by Essler-Korepin-Schoutens. In both cases clear discrete structures appear.
    Preview · Article · Jan 2015 · Journal of Physics A Mathematical and Theoretical
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    Thomas Lam · Pavlo Pylyavskyy · Reiho Sakamoto
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    ABSTRACT: Rigged configurations are known to provide action-angle variables for remarkable discrete dynamical systems known as box-ball systems. We conjecture an explicit piecewise-linear formula to obtain the shapes of a rigged configuration from a tensor product of one-row crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric R-matrix. Our piecewise-linear 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 piecewise-linear formula for the lengths of the solitons of a box-ball system.
    Preview · Article · Oct 2014
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    Anatol N. Kirillov · Reiho Sakamoto
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    ABSTRACT: We compute the energy eigenvalues of Nepomechie--Wang's eigenstates for the spin 1/2 isotropic Heisenberg chain.
    Full-text · Article · Jun 2014 · Moscow Mathematical Journal
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    Anatol N. Kirillov · Reiho Sakamoto
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    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 non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie-Wang. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
    Preview · Article · Feb 2014 · Journal of Physics A Mathematical and Theoretical
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    Reiho Sakamoto
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    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 Kirillov-Reshetikhin crystals and the set of the rigged configurations.
    Preview · Article · Feb 2013 · Symmetry Integrability and Geometry Methods and Applications
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    Thomas Lam · Pavlo Pylyavskyy · Reiho Sakamoto
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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 box-basket-ball system, generalizing the box-ball system of Takahashi and Satsuma. In box-basket-ball 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.
    Preview · Article · Sep 2012 · Reviews in Mathematical Physics
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    Masato Okado · Reiho Sakamoto
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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 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.
    Preview · Article · Nov 2011 · Discrete mathematics & theoretical computer science DMTCS
  • Atsuo Kuniba · Reiho Sakamoto · Yasuhiko Yamada
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    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 anti-particles 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.
    No preview · Article · Oct 2011
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    Masato Okado · Reiho Sakamoto · Anne Schilling
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    ABSTRACT: Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin 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 n-1,n) in terms of tableaux of rectangular shape r x s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.
    Full-text · Article · Sep 2011 · Journal of Algebraic Combinatorics
  • Thomas Lam · Pavlo Pylyavskyy · Reiho Sakamoto

    No preview · Article · Nov 2010
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    Masato Okado · Reiho Sakamoto
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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 Littlewood-Richardson coefficients. Combining this result with math.CO/9901037 and arXiv:1002.3715 we settle the X=M conjecture under the large rank hypothesis.
    Preview · Article · Aug 2010 · Advances in Mathematics
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    Atsuo Kuniba · Reiho Sakamoto · Yasuhiko Yamada
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    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 anti-particles 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"
    Full-text · Article · Jan 2010
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    Anatol N. Kirillov · Reiho Sakamoto
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    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 Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution. Comment: 15 pages, typos corrected, French abstract added, version for publication in proceedings of FPSAC 2010.
    Full-text · Article · Dec 2009
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    Masato Okado · Reiho Sakamoto
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    ABSTRACT: We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov--Reshetikhin crystals $B^{r,k}\otimes B^{1,l}$ of type $D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}$. 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
    Preview · Article · Mar 2009 · International Mathematics Research Notices
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    Anatol N. Kirillov · Reiho Sakamoto
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    ABSTRACT: There are two distinct approaches to the study of initial value problem of the periodic box-ball systems. One way is the rigged configuration approach due to Kuniba--Takagi--Takenouchi and another way is the 10-elimination approach due to Mada--Idzumi--Tokihiro. In this paper, we describe precisely interrelations between these two approaches. Comment: 16 pages, final version, minor revision
    Full-text · Article · Feb 2009 · Letters in Mathematical Physics
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    Anatol N. Kirillov · Reiho Sakamoto
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    ABSTRACT: We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight 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 Haglund--Haiman--Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics. Comment: 35 pages, minor revision, final version
    Preview · Article · Nov 2008 · Moscow Mathematical Journal
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    Atsuo Kuniba · Reiho Sakamoto
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    ABSTRACT: We reformulate the Kerov-Kirillov-Reshetikhin (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 box-ball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function.
    Full-text · Article · Jun 2008 · Reviews in Mathematical Physics
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    Reiho Sakamoto
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    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 box-ball systems formulated by Kuniba-Takagi-Takenouchi (arXiv:math/0602481v2).
    Preview · Article · May 2008
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    Reiho Sakamoto
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    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.
    Preview · Article · Feb 2008 · Journal of Algebraic Combinatorics