The Group Action of the Alternating Group of Order Twelve as Rotations of a Tetrahedron

## 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, and the face adjacent to the three vertices ${v}_{i}$, ${v}_{j}$, ${v}_{k}$, by ${f}_{ijk}$, $i. 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 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 $v1↦ v2and v3↦v4.$ Note that ${S}^{3}=1,$ ${T}^{2}=1,$ and ${\left(ST\right)}^{3}=1.$

Let $P= pij ∣1≤i,j≤4 , V= v1, v2, v3, v4 , E= e12, e13, e14, e23, e24, e34 F= f123, f124, f134, f234 ,and R= r1234 ,$ 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
${\left({A}_{4}\right)}_{{p}_{ij}}=⟨1⟩$ $1$ ${A}_{4}{p}_{ij}=P$ $12$
${\left({A}_{4}\right)}_{{v}_{4}}=\left\{1,S,{S}^{2}\right\}=H$ $3$ ${A}_{4}{v}_{4}=V$ $4$
${\left({A}_{4}\right)}_{{v}_{3}}=\left\{1,TS{T}^{-1},T{S}^{2}{T}^{-1}\right\}=TH{T}^{-1}$ $3$ ${A}_{4}{v}_{3}=V$ $4$
${\left({A}_{4}\right)}_{{v}_{1}}=\left\{1,TS,{S}^{2}T\right\}=\left(ST\right)H{\left(ST\right)}^{-1}$ $3$ ${A}_{4}{v}_{1}=V$ $4$
${\left({A}_{4}\right)}_{{v}_{2}}=\left\{1,ST,{\left(ST\right)}^{2}\right\}=\left({S}^{2}T\right)H{\left({S}^{2}T\right)}^{-1}$ $3$ ${A}_{4}{v}_{3}=V$ $4$
${\left({A}_{4}\right)}_{{e}_{12}}=\left\{1,T\right\}$ $2$ ${A}_{4}{e}_{12}=E$ $6$
${\left({A}_{4}\right)}_{{e}_{34}}=\left\{1,T\right\}$ $2$ ${A}_{4}{e}_{34}=E$ $6$
${\left({A}_{4}\right)}_{{e}_{14}}=\left\{1,ST{S}^{-1}\right\}$ $2$ ${A}_{4}{e}_{14}=E$ $6$
${\left({A}_{4}\right)}_{{e}_{23}}=\left\{1,ST{S}^{-1}\right\}$ $2$ ${A}_{4}{e}_{23}=E$ $6$
${\left({A}_{4}\right)}_{{e}_{13}}=\left\{1,{S}^{2}T{S}^{-2}\right\}$ $2$ ${A}_{4}{e}_{13}=E$ $6$
${\left({A}_{4}\right)}_{{e}_{24}}=\left\{1,{S}^{2}T{S}^{-2}\right\}$ $2$ ${A}_{4}{e}_{24}=E$ $6$
${\left({A}_{4}\right)}_{{f}_{123}}=\left\{1,S,{S}^{2}\right\}$ $3$ ${A}_{4}{f}_{123}=F$ $4$
${\left({A}_{4}\right)}_{{f}_{124}}=\left\{1,TS{T}^{-1},T{S}^{2}{T}^{-1}\right\}$ $3$ ${A}_{4}{f}_{124}=F$ $4$
${\left({A}_{4}\right)}_{{f}_{234}}=\left\{1,\left(ST\right)S{\left(ST\right)}^{-1},\left(ST\right){S}^{2}{\left(ST\right)}^{-1}\right\}$ $3$ ${A}_{4}{f}_{234}=F$ $4$
${\left({A}_{4}\right)}_{{f}_{134}}=\left\{1,\left({S}^{2}T\right)S{\left({S}^{2}T\right)}^{-1},\left({S}^{2}T\right){S}^{2}{\left({S}^{2}T\right)}^{-1}\right\}$ $3$ ${A}_{4}{f}_{134}=F$ $4$
${\left({A}_{4}\right)}_{{r}_{1234}}={A}_{4}$ $12$ ${A}_{4}{r}_{1234}=F$ $1$

 $1$ $v3$ $v2$ $v1$ $v4$ $S$ $v2$ $v1$ $v3$ $v4$ $S2$ $v1$ $v3$ $v2$ $v4$ $T$ $v4$ $v1$ $v2$ $v3$ $ST$ $v1$ $v2$ $v4$ $v3$ $S2T$ $v2$ $v4$ $v1$ $v3$ $TS$ $v4$ $v3$ $v1$ $v2$ $TST$ $v3$ $v1$ $v4$ $v2$ $S2TS$ $v1$ $v4$ $v3$ $v2$ $TS2$ $v4$ $v2$ $v3$ $v1$ $STS2$ $v2$ $v3$ $v4$ $v1$ $S2TS2$ $v3$ $v4$ $v2$ $v1$

## 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)