Affiliation:
1. College of Science and Engineering, Flinders University, Flinders at Tonsley, South Australia 5042, Australia
Abstract
By considering the normalizers of reflection subgroups of types
A
(1)
1
and
A
(1)
3
in
W
~
(
D
5
(
1
)
)
, two subgroups:
W
~
(
A
3
×
A
1
)
(
1
)
⋉
W
(
A
1
(
1
)
)
and
W
~
(
A
1
×
A
1
)
(
1
)
⋉
W
(
A
3
(
1
)
)
can be constructed from a (
A
3
×
A
1
×
A
1
)
(1)
type subroot system. These two symmetries arose in the studies of discrete Painlevé equations (Kajiwara K, Noumi M, Yamada Y. 2002
q
-Painlevé systems arising from
q
-KP hierarchy.
Lett. Math. Phys.
62
, 259–268; Takenawa T. 2003 Weyl group symmetry of type
D
(1)
5
in the
q
-Painlevé V equation.
Funkcial. Ekvac.
46
, 173–186; Okubo N, Suzuki T. 2018 Generalized
q
-Painlevé VI systems of type (
A
2
n
+1
+
A
1
+
A
1
)
(1)
arising from cluster algebra. (
http://arxiv.org/abs/quant-ph/1810.03252
)), where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups (Howlett RB. 1980 Normalizers of parabolic subgroups of reflection groups.
J. London Math. Soc. (2)
21
, 62–80; Brink B, Howlett RB. 1999 Normalizers of parabolic subgroups in Coxeter groups.
Invent. Math.
136
, 323–351). This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.
Subject
General Physics and Astronomy,General Engineering,General Mathematics