The group action of $D_4$ 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 $D_4$ as rotations and reflections of a square

$D_4$ is the group of rotations and reflections of a square. We shall denote the vertices by $v_i$, the edge connecting the vertex $i$ to the vertex $j$ by $e_{ij}$, $i<j$, and the face by $f_{0123}$. Let $p_{ij}$, $0\le i,j\le 2$, denote the point on the edge connecting $v_i$ to $v_j$ which is a third of the way from $v_i$ to $v_j$.

Let $x$ be the $90°$ counterclockwise rotation about the center taking $$v_0\mapsto v_1\mapsto v_2\mapsto v_3\mapsto v_0.$$ Let $y$ be the reflection about the line connecting vertex $v_0$ with vertex $v_2$, taking $$v_1\mapsto v_3\text{and fixing} v_0 \text{and} v_2.$$ Note that $x^4=1,$ $y_2=1,$ and $yx=x^{-1}y.$

Let $$\begin{array}{l}P=\left\{p_{01},p_{10},p_{12},p_{21},p_{23},p_{32},p_{03},p_{30}\right\},\\ V=\left\{v_0,v_1,v_2,v_3\right\},\\ E=\left\{e_{01},e_{12},e_{23},e_{03}\right\},\text{and}\\ F=\left\{f_{0123}\right\},\end{array}$$ denote the sets of points, vertices, edges, and faces respectively. Since $D_4$ acts on the square, $D_4$ acts on each of these sets.

Stabilizer	Size of Stabilizer	Orbit	Size of Orbit
${\left(D_4\right)}_{p_{ij}}=⟨1⟩$	$1$	$D_4{p}_{ij}=P$	$8$
${\left(D_4\right)}_{v_0}=\left\{1,y\right\}=H$	$2$	$D_4{v}_0=V$	$4$
${\left(D_4\right)}_{v_1}=\left\{1,x^{2}y\right\}=xH{x}^{-1}$	$2$	$D_4{v}_1=V$	$4$
${\left(D_4\right)}_{v_2}=\left\{1,y\right\}=H$	$2$	$D_4{v}_2=V$	$4$
${\left(D_4\right)}_{v_3}=\left\{1,x^{2}y\right\}=xH{x}^{-1}$	$2$	$D_4{v}_3=V$	$4$
${\left(D_4\right)}_{e_{01}}=\left\{1,xy\right\}=J$	$2$	$D_4{e}_{01}=E$	$4$
${\left(D_4\right)}_{e_{23}}=\left\{1,xy\right\}=J$	$2$	$D_4{e}_{23}=E$	$4$
${\left(D_4\right)}_{e_{12}}=\left\{1,x^{3}y\right\}=xJ{x}^{-1}$	$2$	$D_4{e}_{12}=E$	$4$
${\left(D_4\right)}_{e_{03}}=\left\{1,x^{3}y\right\}=xJ{x}^{-1}$	$2$	$D_4{e}_{03}=E$	$4$
${\left(D_4\right)}_{f_{012}}=D_4$	$8$	$D_4{f}_{0123}=F$	$1$

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)

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