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Periodicities of T-systems and Y-systems

Published online by Cambridge University Press:  11 January 2016

Rei Inoue ,
Osamu Iyama ,
Atsuo Kuniba ,
Tomoki Nakanishi  and
Junji Suzuki
Rei Inoue
Affiliation:
Faculty of Pharmaceutical Sciences, Suzuka University of Medical Science, Suzuka, 513-8670, Japanreiiy@suzuka-u.ac.jp
Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japaniyama@math.nagoya-u.ac.jp
Atsuo Kuniba
Affiliation:
Institute of Physics, University of Tokyo, Tokyo, 153-8902, Japanatsuo@gokutan.c.c-tokyo.ac.jp
Tomoki Nakanishi
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japannakanisi@math.nagoya-u.ac.jp
Junji Suzuki
Affiliation:
Department of Physics, Faculty of Science, Shizuoka University, Ohya, 836, Japansjsuzuk@ipc.shizuko.ka.ac.jp
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Abstract

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The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 197 , March 2010 , pp. 59 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

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