The group action of $A_4$ as rotations of a tetrahedron

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 February 2011

The group action of $A_4$ as rotations of a tetrahedron

$A_4$ is the group of rotations of the tetrahedron. 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 adjacent to the three vertices $v_i$, $v_j$, $v_k$, by $f_{ijk}$, $i<j<k$. Let $r_{1234}$ denote the region determined by the the inside of the tetrahedron. Let $p_{ij}$, $1\le i,j\le 4$ 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 $S$ be the $60°$ rotation about the bottom face taking $$v_1\mapsto v_2\mapsto v_3\mapsto v_1\text{and fixing} v_4.$$ Let $T$ be the $180°$ rotation about the line connecting the midpoint of the edge $e_{34}$ with the midpoint of the edge $e_{12}$, taking $$v_1\mapsto v_2\text{and}{v}_3\mapsto v_4.$$ Note that $S^3=1,$ $T^2=1,$ and ${\left(ST\right)}^3=1.$

Let $$\begin{array}{l}P=\left\{p_{ij}\mid 1\le i,j\le 4\right\},\\ V=\left\{v_1,v_2,v_3,v_4\right\},\\ E=\left\{e_{12},e_{13},e_{14},e_{23},e_{24},e_{34}\right\}\\ F=\left\{f_{123},f_{124},f_{134},f_{234}\right\},\text{and}\\ R=\left\{r_{1234}\right\},\end{array}$$ denote the sets of points, vertices, edges, faces and regions respectively. Since $A_4$ acts on the tetrahedron, $A_4$ acts on each of these sets.

Stabilizer	Size of Stabilizer	Orbit	Size of Orbit
${\left(A_4\right)}_{p_{ij}}=⟨1⟩$	$1$	$A_4{p}_{ij}=P$	$12$
${\left(A_4\right)}_{v_4}=\left\{1,S,S^2\right\}=H$	$3$	$A_4{v}_4=V$	$4$
${\left(A_4\right)}_{v_3}=\left\{1,TS{T}^{-1},T{S}^2{T}^{-1}\right\}=TH{T}^{-1}$	$3$	$A_4{v}_3=V$	$4$
${\left(A_4\right)}_{v_1}=\left\{1,TS,S^{2}T\right\}=\left(ST\right)H{\left(ST\right)}^{-1}$	$3$	$A_4{v}_1=V$	$4$
${\left(A_4\right)}_{v_2}=\left\{1,ST,{\left(ST\right)}^2\right\}=\left(S^{2}T\right)H{\left(S^{2}T\right)}^{-1}$	$3$	$A_4{v}_3=V$	$4$
${\left(A_4\right)}_{e_{12}}=\left\{1,T\right\}$	$2$	$A_4{e}_{12}=E$	$6$
${\left(A_4\right)}_{e_{34}}=\left\{1,T\right\}$	$2$	$A_4{e}_{34}=E$	$6$
${\left(A_4\right)}_{e_{14}}=\left\{1,ST{S}^{-1}\right\}$	$2$	$A_4{e}_{14}=E$	$6$
${\left(A_4\right)}_{e_{23}}=\left\{1,ST{S}^{-1}\right\}$	$2$	$A_4{e}_{23}=E$	$6$
${\left(A_4\right)}_{e_{13}}=\left\{1,S^{2}T{S}^{-2}\right\}$	$2$	$A_4{e}_{13}=E$	$6$
${\left(A_4\right)}_{e_{24}}=\left\{1,S^{2}T{S}^{-2}\right\}$	$2$	$A_4{e}_{24}=E$	$6$
${\left(A_4\right)}_{f_{123}}=\left\{1,S,S^2\right\}$	$3$	$A_4{f}_{123}=F$	$4$
${\left(A_4\right)}_{f_{124}}=\left\{1,TS{T}^{-1},T{S}^2{T}^{-1}\right\}$	$3$	$A_4{f}_{124}=F$	$4$
${\left(A_4\right)}_{f_{234}}=\left\{1,\left(ST\right)S{\left(ST\right)}^{-1},\left(ST\right)S^2{\left(ST\right)}^{-1}\right\}$	$3$	$A_4{f}_{234}=F$	$4$
${\left(A_4\right)}_{f_{134}}=\left\{1,\left(S^{2}T\right)S{\left(S^{2}T\right)}^{-1},\left(S^{2}T\right)S^2{\left(S^{2}T\right)}^{-1}\right\}$	$3$	$A_4{f}_{134}=F$	$4$
${\left(A_4\right)}_{r_{1234}}=A_4$	$12$	$A_4{r}_{1234}=F$	$1$

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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