Geometric Reductions of ABS equations on an n-cube to discrete Painlevé systems

Nalini Joshi, Nobutaka Nakazono, Yang Shi

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the four-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type A 5 (1) with the extended affine Weyl group W̃ ((A 2 + A 1 ) (1) ). We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.

Original languageEnglish
Article number505201
Pages (from-to)505201-505216
Number of pages16
JournalJournal of Physics. A. Mathematical and Theoretical
Issue number50
Publication statusPublished - 25 Nov 2014
Externally publishedYes


  • geometric reductionPACS numbers: 02.30.Ik, 02.20.Qs
  • discrete Painlevé equations
  • ABS equations
  • Geometric reduction
  • Discrete Painlevé equations


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