Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 027, 18 pages      arXiv:0906.1410      https://doi.org/10.3842/SIGMA.2010.027
Contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems”

Level Set Structure of an Integrable Cellular Automaton

Taichiro Takagi
Department of Applied Physics, National Defense Academy, Kanagawa 239-8686, Japan

Received October 23, 2009, in final form March 15, 2010; Published online March 31, 2010

Abstract
Based on a group theoretical setting a sort of discrete dynamical system is constructed and applied to a combinatorial dynamical system defined on the set of certain Bethe ansatz related objects known as the rigged configurations. This system is then used to study a one-dimensional periodic cellular automaton related to discrete Toda lattice. It is shown for the first time that the level set of this cellular automaton is decomposed into connected components and every such component is a torus.

Key words: periodic box-ball system; rigged configuration; invariant torus.

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References

  1. Arnold V.I., Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1989.
  2. Yura F., Tokihiro T., On a periodic soliton cellular automaton, J. Phys. A: Math. Gen. 35 (2002), 3787-3801, nlin.SI/0112041.
  3. Yoshihara D., Yura F., Tokihiro T., Fundamental cycle of a periodic box-ball system, J. Phys. A: Math. Gen. 36 (2003), 99-121, nlin.SI/0208042.
  4. Iwao S., Tokihiro T., Ultradiscretization of the theta function solution of pd Toda, J. Phys. A: Math. Gen. 40 (2007), 12987-13021, arXiv:0705.4013.
  5. Kuniba A., Takagi T., Takenouchi A., Bethe ansatz and inverse scattering transform in a periodic box-ball system, Nuclear Phys. B 747 (2006), 354-397, math.QA/0602481.
  6. Inoue R., Takenawa T., Tropical spectral curves and integrable cellular automata, Int. Math. Res. Not. 2008 (2008), Art ID. rnn019, 27 pages, arXiv:0704.2471.
  7. Kuniba A., Sakamoto R., Bethe ansatz in a periodic box-ball system and ultradiscrete Riemann theta function, J. Stat. Mech. Theory Exp. 2006 (2006), no. 9, P09005, 12 pages, math.QA/0606208.
  8. Kuniba A., Nakanishi T., The Bethe equation at q=0, the Möbius inversion formula, and weight multiplicities. I. The sl(2) case, in Physical Combinatorics (Kyoto, 1999), Progr. Math., Vol. 191, Birkhäuser Boston, Boston, MA, 2000, 185-216, math.QA/9909056.
  9. Kerov S.V., Kirillov A.N., Reshetikhin N.Yu., Combinatorics, Bethe ansatz, and representations of the symmetric group, Zap. Nauch. Semin. LOMI 155 (1986), 50-64 (English transl.: J. Soviet Math. 41 (1988), 916-924).
  10. Kirillov A.N., Reshetikhin N.Yu., The Bethe ansatz and the combinatorics of Young tableaux, Zap. Nauch. Semin. LOMI 155 (1986), 65-115 (English transl.: J. Soviet Math. 41 (1988), 925-955).
  11. Kashiwara M., Crystal bases of modified quantized universal enveloping algebra, Duke Math. J. 73 (1994), 383-413.
  12. Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.
  13. Kuniba A., Takagi T., Bethe ansatz, inverse scattering transform and tropical Riemann theta function in a periodic soliton cellular automaton for An(1), SIGMA 6 (2010), 013, 52 pages, arXiv:0909.3759.
  14. Stanley R.P., Enumerative Combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge University Press, Cambridge, 1997.
  15. Takagi T., Soliton cellular automata, in Combinatorial Aspect of Integrable Systems, MSJ Mem., Vol. 17, Math. Soc. Japan, Tokyo, 2007, 105-144.

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