T-systems with boundaries from network solutions

Research output: Contribution to journalArticle

Abstract

In this paper, we use the network solution of the Ar T-system to derive that of the unrestricted A T -system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted T-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for T-systems (corresponding to the case A × Ar) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the T-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

Original languageEnglish (US)
JournalElectronic Journal of Combinatorics
Volume20
Issue number1
StatePublished - Jan 7 2013

Fingerprint

Algebra
Boundary conditions
Pentagram
Octahedron
Cluster Algebra
Periodicity
Torus
Symmetry

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

T-systems with boundaries from network solutions. / Di Francesco, Philippe; Kedem, Rinat.

In: Electronic Journal of Combinatorics, Vol. 20, No. 1, 07.01.2013.

Research output: Contribution to journalArticle

@article{0ccc7f5c9de441338b390511993d3cf0,
title = "T-systems with boundaries from network solutions",
abstract = "In this paper, we use the network solution of the Ar T-system to derive that of the unrestricted A∞ T -system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted T-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for T-systems (corresponding to the case Aℓ × Ar) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the T-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.",
author = "{Di Francesco}, Philippe and Rinat Kedem",
year = "2013",
month = "1",
day = "7",
language = "English (US)",
volume = "20",
journal = "Electronic Journal of Combinatorics",
issn = "1077-8926",
publisher = "Electronic Journal of Combinatorics",
number = "1",

}

TY - JOUR

T1 - T-systems with boundaries from network solutions

AU - Di Francesco, Philippe

AU - Kedem, Rinat

PY - 2013/1/7

Y1 - 2013/1/7

N2 - In this paper, we use the network solution of the Ar T-system to derive that of the unrestricted A∞ T -system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted T-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for T-systems (corresponding to the case Aℓ × Ar) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the T-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

AB - In this paper, we use the network solution of the Ar T-system to derive that of the unrestricted A∞ T -system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted T-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for T-systems (corresponding to the case Aℓ × Ar) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the T-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

UR - http://www.scopus.com/inward/record.url?scp=84872741424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872741424&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84872741424

VL - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -