scholarly journals Affine Weyl Group Symmetry of the Garnier System

2005 ◽  
Vol 48 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Takao Suzuki
Keyword(s):  
Weyl Group ◽  
Group Symmetry ◽  
Affine Weyl Group ◽  
Garnier System

scholarly journals Lax pairs of discrete Painlevé equations: ( A 2 + A 1 ) (1) case

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160696 ◽  
Author(s):  
Nalini Joshi ◽  
Nobutaka Nakazono
Keyword(s):  
Weyl Group ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Lax Pairs ◽  
Painleve Equations ◽  
Comprehensive Method ◽  
Affine Weyl Group ◽  
Discrete Painlevé Equations

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the A 5 ( 1 ) -surface q -Painlevé system, which has the affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . Two new Lax pairs are found.

scholarly journals Generalized q-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

2020 ◽  
Author(s):  
Naoto Okubo ◽  
Takao Suzuki
Keyword(s):  
Weyl Group ◽  
Cluster Algebra ◽  
Higher Order ◽  
Root Lattice ◽  
Similarity Reduction ◽  
Birational Transformations ◽  
System A ◽  
Affine Weyl Group ◽  
Painlevé Vi Equation ◽  
Garnier System

Abstract In this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

The affine Weyl group and modular invariant partition functions

1988 ◽  
Vol 205 (2-3) ◽  
pp. 281-284 ◽  
Author(s):  
D. Altschüler ◽  
J. Lacki ◽  
Ph. Zaugg
Keyword(s):  
Weyl Group ◽  
Partition Functions ◽  
Modular Invariant ◽  
Affine Weyl Group
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