Periodicities of T-systems and Y-systems

By Rei Inoue, Osamu Iyama, A. Kuniba, Tomoki Nakanishi, J. Suzuki. Nagoya mathematical journal, 2008

83 citations recorded by the source corpus.

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).

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/76e90fb16fd78047de239eb0fa2cbde611f525d5

How researchers see this paper

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In Algebra and Number Theory

1. B. Keller
2. A. Kuniba
3. Tomoki Nakanishi
4. J. Suzuki
5. H. Nakajima
6. Bernard Leclerc
7. D. Hernandez
8. I. Reiten

In Geometry and Topology

1. B. Keller
2. Bernard Leclerc
3. Tomoki Nakanishi
4. A. Kuniba
5. J. Suzuki
6. H. Nakajima
7. Pierre-Guy Plamondon
8. D. Hernandez

In Statistical and Nonlinear Physics

1. B. Keller
2. H. Nakajima
3. Tomoki Nakanishi
4. A. Kuniba
5. J. Suzuki
6. Bernard Leclerc
7. C. Geiss
8. J. Schröer

Citations in the corpus

• Quiver varieties and cluster algebras (2009, background): “Graphically it is given by Ω̃ : i s %%K KK KK K r // j k t 99rrrrrr =⇒ Ω̃∗ : i r+st // j tyyrr rr rr k s eeKKKKKK , where s, t are nonnegative integers and i l−→ j means that there are l arrows from i to j if l ≥ 0, (−l) arrows from j to i if l ≤ 0.”
• R-Twisting and 4d/2d Correspondences (2010, unknown): “[80, 81, 89–91] and references therein).”
• Cluster algebras and derived categories (2012, unknown): “• discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] .”
• The periodicity conjecture for pairs of Dynkin diagrams (2010, methodology): “We nevertheless define them since they are useful in other applications of the categorical model that we will construct, notably in the study [39] of T -systems.”
• T-systems and Y-systems in integrable systems (2010, methodology): “Casoratian solutions are known also for the restricted T-systems for Ar [124] and Cr [17].”