Last updates: 17 December 2010

${D}_{4}$ 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 $90\xb0$ 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 $${v}_{1}\mapsto {v}_{3}\phantom{\rule{3em}{0ex}}\text{and fixing}{v}_{0}\text{and}{v}_{2}.$$ Note that ${x}^{4}=1,$ ${y}_{2}=1,$ and $yx={x}^{-1}y.$

Let $$\begin{array}{l}P=\left\{{p}_{01},{p}_{10},{p}_{12},{p}_{21},{p}_{23},{p}_{32},{p}_{03},{p}_{30}\right\},\\ V=\left\{{v}_{0},{v}_{1},{v}_{2},{v}_{3}\right\},\\ E=\left\{{e}_{01},{e}_{12},{e}_{23},{e}_{03}\right\},\phantom{\rule{3em}{0ex}}\text{and}\\ F=\left\{{f}_{0123}\right\},\end{array}$$ 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 |
---|---|---|---|

${\left({D}_{4}\right)}_{{p}_{ij}}=\u27e81\u27e9$ | $1$ | ${D}_{4}{p}_{ij}=P$ | $8$ |

${\left({D}_{4}\right)}_{{v}_{0}}=\left\{1,y\right\}=H$ | $2$ | ${D}_{4}{v}_{0}=V$ | $4$ |

${\left({D}_{4}\right)}_{{v}_{1}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ | $2$ | ${D}_{4}{v}_{1}=V$ | $4$ |

${\left({D}_{4}\right)}_{{v}_{2}}=\left\{1,y\right\}=H$ | $2$ | ${D}_{4}{v}_{2}=V$ | $4$ |

${\left({D}_{4}\right)}_{{v}_{3}}=\left\{1,{x}^{2}y\right\}=xH{x}^{-1}$ | $2$ | ${D}_{4}{v}_{3}=V$ | $4$ |

${\left({D}_{4}\right)}_{{e}_{01}}=\left\{1,xy\right\}=J$ | $2$ | ${D}_{4}{e}_{01}=E$ | $4$ |

${\left({D}_{4}\right)}_{{e}_{23}}=\left\{1,xy\right\}=J$ | $2$ | ${D}_{4}{e}_{23}=E$ | $4$ |

${\left({D}_{4}\right)}_{{e}_{12}}=\left\{1,{x}^{3}y\right\}=xJ{x}^{-1}$ | $2$ | ${D}_{4}{e}_{12}=E$ | $4$ |

${\left({D}_{4}\right)}_{{e}_{03}}=\left\{1,{x}^{3}y\right\}=xJ{x}^{-1}$ | $2$ | ${D}_{4}{e}_{03}=E$ | $4$ |

${\left({D}_{4}\right)}_{{f}_{012}}={D}_{4}$ | $8$ | ${D}_{4}{f}_{0123}=F$ | $1$ |

[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)