The Group Action of the Dihedral Group of Order Six as Symmetries of an Equilateral Triangle

## The group action of ${D}_{3}$ as rotations and reflections of an equilateral triangle

${D}_{3}$ is the group of rotations and reflections of an equilateral triangle. 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 ${f}_{012}$. 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 $120°$ counterclockwise rotation about the center taking $v0↦ v1↦ v2↦ v0.$ Let $y$ be the reflection about the line connecting vertex ${v}_{0}$ with the midpoint of the edge ${e}_{12}$, taking Note that ${x}^{3}=1,$ ${y}_{2}=1,$ and $yx={x}^{-1}y.$

Let $P= p01, p10, p12, p21, p02, p20 , V= v0, v1, v2 , E= e01, e12, e02 ,and F= f012 ,$ denote the sets of points, vertices, edges, and faces respectively. Since ${D}_{3}$ acts on the equilateral triangle, ${D}_{3}$ acts on each of these sets.

Stabilizer Size of Stabilizer Orbit Size of Orbit
${\left({D}_{3}\right)}_{{p}_{ij}}=⟨1⟩$ $1$ ${D}_{3}{P}_{ij}=P$ $6$
${\left({D}_{3}\right)}_{{v}_{0}}=\left\{1,y\right\}=H$ $2$ ${D}_{3}{v}_{0}=V$ $3$
${\left({D}_{3}\right)}_{{v}_{1}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ $2$ ${D}_{3}{v}_{1}=V$ $3$
${\left({D}_{3}\right)}_{{v}_{2}}=\left\{1,xy\right\}={x}^{2}H{x}^{-2}$ $2$ ${D}_{3}{v}_{2}=V$ $3$
${\left({D}_{3}\right)}_{{e}_{01}}=\left\{1,xy\right\}={x}^{2}H{x}^{-2}$ $2$ ${D}_{3}{e}_{01}=E$ $3$
${\left({D}_{3}\right)}_{{e}_{12}}=\left\{1,y\right\}=H$ $2$ ${D}_{3}{e}_{12}=E$ $3$
${\left({D}_{3}\right)}_{{e}_{02}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ $2$ ${D}_{3}{e}_{02}=E$ $3$
${\left({D}_{3}\right)}_{{f}_{012}}={D}_{3}$ $6$ ${D}_{3}{f}_{012}=F$ $1$

 $v0$ $v2$ $v1$ $1$ $v2$ $v1$ $v0$ $x$ $v1$ $v0$ $v2$ $x2$ $v0$ $v1$ $v2$ $y$ $v1$ $v2$ $v0$ $xy$ $v2$ $v0$ $v1$ ${x}^{2}y$

## References

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[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)