Reiho Sakamoto

Tokyo University of Science, Edo, Tōkyō, Japan

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Publications (26)32.55 Total impact

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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.
    10/2014;
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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.
    06/2014;
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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.
    Journal of Physics A Mathematical and Theoretical 02/2014; 47(20). · 1.77 Impact Factor
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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.
    Symmetry Integrability and Geometry Methods and Applications 02/2013; · 1.30 Impact Factor
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    Reiho Sakamoto
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    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;
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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.
    Reviews in Mathematical Physics 09/2012; 24(8):50019-. · 1.45 Impact Factor
  • 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.
    10/2011;
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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.
    Journal of Algebraic Combinatorics 09/2011; · 0.72 Impact Factor
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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.
    Advances in Mathematics 08/2010; · 1.35 Impact Factor
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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"
    01/2010;
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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.
    12/2009;
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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
    International Mathematics Research Notices 03/2009; · 1.07 Impact Factor
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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
    Letters in Mathematical Physics 02/2009; · 2.07 Impact Factor
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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
    Moscow Mathematical Journal 11/2008; · 0.35 Impact Factor
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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.
    Reviews in Mathematical Physics 06/2008; 20(05):493-527. · 1.45 Impact Factor
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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).
    05/2008;
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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.
    Journal of Algebraic Combinatorics 01/2008; 27(1):55-98. · 0.72 Impact Factor
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    Reiho Sakamoto
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    ABSTRACT: The Kirillov-Schilling-Shimozono (KSS) bijection appearing in theory of the Fermionic formula gives one to one correspondence between the set of elements of tensor products of the Kirillov-Reshetikhin crystals (called paths) and the set of rigged configurations. It is generalization of Kerov-Kirillov-Reshetikhin bijection and plays inverse scattering formalism for the box-ball 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
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    Atsuo Kuniba, Reiho Sakamoto, Yasuhiko Yamada
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    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 N-soliton solutions of the box–ball system.
    Nuclear Physics B 12/2007; · 3.95 Impact Factor
  • Source
    Reiho Sakamoto
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    ABSTRACT: The Kerov--Kirillov--Reshetikhin (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 box-ball systems.
    Journal of Physics A Mathematical and Theoretical 09/2007; · 1.77 Impact Factor

Publication Stats

145 Citations
32.55 Total Impact Points

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