The group action of D4 as rotations and reflections of a square
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 17 December 2010
The group action of D4 as rotations and reflections of a square
D4 is the group of rotations and reflections of a square. We shall denote the vertices by
vi, the edge connecting the vertex
i to the vertex
j by
eij, i<j, and the face by
f0123. Let
pij, 0≤i,j≤2, denote the point on the edge connecting vi to vj which is a third of the way from vi to vj.
Let x be the 90° counterclockwise rotation about the center taking
v0↦v1↦v2↦v3↦v0.
Let y be the reflection about the line connecting vertex v0 with vertex
v2, taking
v1↦v3and fixingv0andv2.
Note that
x4=1,y2=1,
and
yx=x-1y.
Let
P=p01,p10,p12,p21,p23,p32,p03,p30,V=v0,v1,v2,v3,E=e01,e12,e23,e03,andF=f0123,
denote the sets of points, vertices, edges, and faces respectively. Since D4 acts on the square, D4 acts on each of these sets.
Stabilizer
Size of Stabilizer
Orbit
Size of Orbit
D4pij=⟨1⟩
1
D4pij=P
8
D4v0=1,y=H
2
D4v0=V
4
D4v1=1,x2y=xHx-1
2
D4v1=V
4
D4v2=1,y=H
2
D4v2=V
4
D4v3=1,x2y=xHx-1
2
D4v3=V
4
D4e01=1,xy=J
2
D4e01=E
4
D4e23=1,xy=J
2
D4e23=E
4
D4e12=1,x3y=xJx-1
2
D4e12=E
4
D4e03=1,x3y=xJx-1
2
D4e03=E
4
D4f012=D4
8
D4f0123=F
1
References
[CM]
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups,
Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980.
MR0562913 (81a:20001)
[GW1]
F. Goodman and H. Wenzl,
The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334.
MR1242201 (95c:16020)