The Group Action of the Dihedral Group of Order Eight as Symmetries of a Square

## 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, 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 $v0↦ v1↦ v2↦ v3↦ v0.$ Let $y$ be the reflection about the line connecting vertex ${v}_{0}$ with vertex ${v}_{2}$, taking Note that ${x}^{4}=1,$ ${y}_{2}=1,$ and $yx={x}^{-1}y.$

Let $P= p01, p10, p12, p21, p23, p32, p03, p30 , V= v0, v1, v2, v3 , E= e01, e12, e23, e03 ,and F= f0123 ,$ 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
${\left({D}_{4}\right)}_{{p}_{ij}}=⟨1⟩$ $1$ ${D}_{4}{p}_{ij}=P$ $8$
${\left({D}_{4}\right)}_{{v}_{0}}=\left\{1,y\right\}=H$ $2$ ${D}_{4}{v}_{0}=V$ $4$
${\left({D}_{4}\right)}_{{v}_{1}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ $2$ ${D}_{4}{v}_{1}=V$ $4$
${\left({D}_{4}\right)}_{{v}_{2}}=\left\{1,y\right\}=H$ $2$ ${D}_{4}{v}_{2}=V$ $4$
${\left({D}_{4}\right)}_{{v}_{3}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ $2$ ${D}_{4}{v}_{3}=V$ $4$
${\left({D}_{4}\right)}_{{e}_{01}}=\left\{1,xy\right\}=J$ $2$ ${D}_{4}{e}_{01}=E$ $4$
${\left({D}_{4}\right)}_{{e}_{23}}=\left\{1,xy\right\}=J$ $2$ ${D}_{4}{e}_{23}=E$ $4$
${\left({D}_{4}\right)}_{{e}_{12}}=\left\{1,{x}^{3}y\right\}=xJ{x}^{-1}$ $2$ ${D}_{4}{e}_{12}=E$ $4$
${\left({D}_{4}\right)}_{{e}_{03}}=\left\{1,{x}^{3}y\right\}=xJ{x}^{-1}$ $2$ ${D}_{4}{e}_{03}=E$ $4$
${\left({D}_{4}\right)}_{{f}_{012}}={D}_{4}$ $8$ ${D}_{4}{f}_{0123}=F$ $1$

 $1$ $v0$ $v3$ $v2$ $v1$ $x$ $v3$ $v2$ $v1$ $v0$ $x2$ $v2$ $v1$ $v0$ $v3$ $x3$ $v1$ $v0$ $v3$ $v2$ $y$ $v0$ $v1$ $v2$ $v3$ $xy$ $v1$ $v2$ $v3$ $v0$ $x2y$ $v2$ $v3$ $v0$ $v1$ $x3y$ $v3$ $v0$ $v1$ $v2$

## 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)