The group action of $S_4$ as rotations of a cube

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

Last updates: 14 February 2011

The group action of $S_4$ as rotations of a cube

$S_4$ is the group of rotations of the cube. 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 four vertices $v_i$, $v_j$, $v_k$, $v_l$ by $f_{ijkl}$, $i<j<k<l$. Let $r_{12345678}$ denote the region determined by the the inside of the cube. Let $p_{ij}$ 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 $90°$ rotation about the top face taking $$\begin{array}{ll}{v}_1\mapsto v_2\mapsto v_3\mapsto v_4\mapsto v_1& \text{and}\\ v_5\mapsto v_6\mapsto v_7\mapsto v_8\mapsto v_5.\end{array}$$ Let $T$ be the $90°$ rotation about the right face taking $$\begin{array}{ll}{v}_4\mapsto v_1\mapsto v_5\mapsto v_8\mapsto v_1& \text{and}\\ v_3\mapsto v_2\mapsto v_6\mapsto v_7\mapsto v_3.\end{array}$$ Note that $S^4=1,$ $T^4=1,$ and ${\left(ST\right)}^3=1.$

Let $$\begin{array}{l}P=\left\{p_{ij}\mid 1\le i,j\le 8\right\},\\ V=\left\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right\},\\ E=\left\{e_{12},e_{23},e_{34},e_{13},e_{15},e_{48},e_{26},e_{37},e_{56},e_{67},e_{78},e_{58}\right\}\\ F=\left\{f_{1234},f_{5678},f_{1256},f_{3478},f_{1458},f_{2367}\right\},\text{and}\\ R=\left\{r_{12345678}\right\},\end{array}$$ denote the sets of points, vertices, edges, faces and regions respectively. Since $S_4$ acts on the cube, $S_4$ acts on each of these sets.

Stabilizer	Size of Stabilizer	Orbit	Size of Orbit
${\left(S_4\right)}_{p_{ij}}=⟨1⟩$	$1$	$S_4{p}_{ij}=P$	$24$
${\left(S_4\right)}_{v_1}=\left\{1,T^{3}S,T{S}^3\right\}=H$	$3$	$S_4{v}_1=V$	$8$
${\left(S_4\right)}_{v_7}=\left\{1,T^{3}S,T{S}^3\right\}=H$	$3$	$S_4{v}_7=V$	$8$
${\left(S_4\right)}_{v_2}=\left\{1,S^3{T}^3,TS\right\}=SH{S}^{-1}$	$3$	$S_4{v}_2=V$	$8$
${\left(S_4\right)}_{v_8}=\left\{1,S^3{T}^3,TS\right\}=SH{S}^{-1}$	$3$	$S_4{v}_8=V$	$8$
${\left(S_4\right)}_{v_8}=\left\{1,S^3{T}^3,TS\right\}=S^{2}H{S}^{-2}$	$3$	$S_4{v}_3=V$	$8$
${\left(S_4\right)}_{v_5}=\left\{1,ST,S^{2}TT\right\}=S^{2}H{S}^{-2}$	$3$	$S_4{v}_5=V$	$8$
${\left(S_4\right)}_{v_4}=\left\{1,S^{3}T,S^{2}T{S}^3\right\}=S^{3}H{S}^{-3}$	$3$	$S_4{v}_4=V$	$8$
${\left(S_4\right)}_{v_6}=\left\{1,S^{3}T,S^{2}T{S}^3\right\}=S^{3}H{S}^{-3}$	$3$	$S_4{v}_6=V$	$8$
${\left(S_4\right)}_{e_{12}}=\left\{1,T{S}^2\right\}=J$	$2$	$S_4{e}_{12}=E$	$12$
${\left(S_4\right)}_{e_{78}}=\left\{1,T{S}^2\right\}=J$	$2$	$S_4{e}_{78}=E$	$12$
${\left(S_4\right)}_{e_{23}}=\left\{1,STS\right\}=SJ{S}^{-1}$	$2$	$S_4{e}_{23}=E$	$12$
${\left(S_4\right)}_{e_{58}}=\left\{1,STS\right\}=SJ{S}^{-1}$	$2$	$S_4{e}_{58}=E$	$12$
${\left(S_4\right)}_{e_{34}}=\left\{1,S^{2}T\right\}=S^{2}J{S}^{-2}$	$2$	$S_4{e}_{34}=E$	$12$
${\left(S_4\right)}_{e_{56}}=\left\{1,S^{2}T\right\}=S^{2}J{S}^{-2}$	$2$	$S_4{e}_{56}=E$	$12$
${\left(S_4\right)}_{e_{14}}=\left\{1,S^{3}T{S}^3\right\}=S^{3}J{S}^{-3}$	$2$	$S_4{e}_{14}=E$	$12$
${\left(S_4\right)}_{e_{67}}=\left\{1,S^{3}T{S}^3\right\}=S^{3}J{S}^{-3}$	$2$	$S_4{e}_{67}=E$	$12$
${\left(S_4\right)}_{e_{15}}=\left\{1,S{T}^3\right\}=\left(S{T}^3\right)J{\left(S{T}^3\right)}^{-1}$	$2$	$S_4{e}_{15}=E$	$12$
${\left(S_4\right)}_{e_{37}}=\left\{1,S{T}^2\right\}=\left(S{T}^3\right)J{\left(S{T}^3\right)}^{-1}$	$2$	$S_4{e}_{37}=E$	$12$
${\left(S_4\right)}_{e_{48}}=\left\{1,S^3{T}^2\right\}=\left(S^{3}TS\right)J{\left(S^{3}TS\right)}^{-1}$	$2$	$S_4{e}_{48}=E$	$12$
${\left(S_4\right)}_{e_{26}}=\left\{1,S^3{T}^2\right\}=\left(S^{3}TS\right)J{\left(S^{3}TS\right)}^{-1}$	$2$	$S_4{e}_{26}=E$	$12$
${\left(S_4\right)}_{f_{1234}}=\left\{1,S,S^2,S^3\right\}=K$	$4$	$S_4{f}_{1234}=F$	$6$
${\left(S_4\right)}_{f_{5678}}=\left\{1,S,S^2,S^3\right\}=K$	$4$	$S_4{f}_{5678}=F$	$6$
${\left(S_4\right)}_{f_{1256}}=\left\{1,S^2{T}^2,S^3{T}^{2}S,ST{S}^3\right\}=TK{T}^{-1}$	$4$	$S_4{f}_{234}=F$	$6$
${\left(S_4\right)}_{f_{3478}}=\left\{1,S^2{T}^2,S^3{T}^{2}S,ST{S}^3\right\}=TK{T}^{-1}$	$4$	$S_4{f}_{3478}=F$	$6$
${\left(S_4\right)}_{f_{1458}}=\left\{1,T,T^2,T^3\right\}=\left(S{T}^3\right)K{\left(S{T}^3\right)}^{-1}$	$4$	$S_4{f}_{1458}=F$	$6$
${\left(S_4\right)}_{f_{2367}}=\left\{1,T,T^2,T^3\right\}=\left(S{T}^3\right)K{\left(S{T}^3\right)}^{-1}$	$4$	$S_4{f}_{2367}=F$	$6$
${\left(S_4\right)}_{r_{12345678}}=S_4$	$24$	$S_4{r}_{12345678}=F$	$1$

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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