The Group Action of the Symmetric Group of Order Twenty Four as Rotations of a Cube

## 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, and the face adjacent to the four vertices ${v}_{i}$, ${v}_{j}$, ${v}_{k}$, ${v}_{l}$ by ${f}_{ijkl}$, $i. 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 $v1↦ v2↦ v3↦ v4↦ v1 and v5↦ v6↦ v7↦ v8↦ v5.$ Let $T$ be the $90°$ rotation about the right face taking $v4↦ v1↦ v5↦ v8↦ v1 and v3↦ v2↦ v6↦ v7↦ v3.$ Note that ${S}^{4}=1,$ ${T}^{4}=1,$ and ${\left(ST\right)}^{3}=1.$

Let $P= pij ∣1≤i,j≤8 , V= v1, v2, v3, v4, v5, v6, v7, v8 , E= e12, e23, e34, e13, e15, e48, e26, e37, e56, e67, e78, e58 F= f1234, f5678, f1256, f3478, f1458, f2367 ,and R= r12345678 ,$ 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
${\left({S}_{4}\right)}_{{p}_{ij}}=⟨1⟩$ $1$ ${S}_{4}{p}_{ij}=P$ $24$
${\left({S}_{4}\right)}_{{v}_{1}}=\left\{1,{T}^{3}S,T{S}^{3}\right\}=H$ $3$ ${S}_{4}{v}_{1}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{7}}=\left\{1,{T}^{3}S,T{S}^{3}\right\}=H$ $3$ ${S}_{4}{v}_{7}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{2}}=\left\{1,{S}^{3}{T}^{3},TS\right\}=SH{S}^{-1}$ $3$ ${S}_{4}{v}_{2}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{8}}=\left\{1,{S}^{3}{T}^{3},TS\right\}=SH{S}^{-1}$ $3$ ${S}_{4}{v}_{8}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{8}}=\left\{1,{S}^{3}{T}^{3},TS\right\}={S}^{2}H{S}^{-2}$ $3$ ${S}_{4}{v}_{3}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{5}}=\left\{1,ST,{S}^{2}TT\right\}={S}^{2}H{S}^{-2}$ $3$ ${S}_{4}{v}_{5}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{4}}=\left\{1,{S}^{3}T,{S}^{2}T{S}^{3}\right\}={S}^{3}H{S}^{-3}$ $3$ ${S}_{4}{v}_{4}=V$ $8$
${\left({S}_{4}\right)}_{{v}_{6}}=\left\{1,{S}^{3}T,{S}^{2}T{S}^{3}\right\}={S}^{3}H{S}^{-3}$ $3$ ${S}_{4}{v}_{6}=V$ $8$
${\left({S}_{4}\right)}_{{e}_{12}}=\left\{1,T{S}^{2}\right\}=J$ $2$ ${S}_{4}{e}_{12}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{78}}=\left\{1,T{S}^{2}\right\}=J$ $2$ ${S}_{4}{e}_{78}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{23}}=\left\{1,STS\right\}=SJ{S}^{-1}$ $2$ ${S}_{4}{e}_{23}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{58}}=\left\{1,STS\right\}=SJ{S}^{-1}$ $2$ ${S}_{4}{e}_{58}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{34}}=\left\{1,{S}^{2}T\right\}={S}^{2}J{S}^{-2}$ $2$ ${S}_{4}{e}_{34}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{56}}=\left\{1,{S}^{2}T\right\}={S}^{2}J{S}^{-2}$ $2$ ${S}_{4}{e}_{56}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{14}}=\left\{1,{S}^{3}T{S}^{3}\right\}={S}^{3}J{S}^{-3}$ $2$ ${S}_{4}{e}_{14}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{67}}=\left\{1,{S}^{3}T{S}^{3}\right\}={S}^{3}J{S}^{-3}$ $2$ ${S}_{4}{e}_{67}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{15}}=\left\{1,S{T}^{3}\right\}=\left(S{T}^{3}\right)J{\left(S{T}^{3}\right)}^{-1}$ $2$ ${S}_{4}{e}_{15}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{37}}=\left\{1,S{T}^{2}\right\}=\left(S{T}^{3}\right)J{\left(S{T}^{3}\right)}^{-1}$ $2$ ${S}_{4}{e}_{37}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{48}}=\left\{1,{S}^{3}{T}^{2}\right\}=\left({S}^{3}TS\right)J{\left({S}^{3}TS\right)}^{-1}$ $2$ ${S}_{4}{e}_{48}=E$ $12$
${\left({S}_{4}\right)}_{{e}_{26}}=\left\{1,{S}^{3}{T}^{2}\right\}=\left({S}^{3}TS\right)J{\left({S}^{3}TS\right)}^{-1}$ $2$ ${S}_{4}{e}_{26}=E$ $12$
${\left({S}_{4}\right)}_{{f}_{1234}}=\left\{1,S,{S}^{2},{S}^{3}\right\}=K$ $4$ ${S}_{4}{f}_{1234}=F$ $6$
${\left({S}_{4}\right)}_{{f}_{5678}}=\left\{1,S,{S}^{2},{S}^{3}\right\}=K$ $4$ ${S}_{4}{f}_{5678}=F$ $6$
${\left({S}_{4}\right)}_{{f}_{1256}}=\left\{1,{S}^{2}{T}^{2},{S}^{3}{T}^{2}S,ST{S}^{3}\right\}=TK{T}^{-1}$ $4$ ${S}_{4}{f}_{234}=F$ $6$
${\left({S}_{4}\right)}_{{f}_{3478}}=\left\{1,{S}^{2}{T}^{2},{S}^{3}{T}^{2}S,ST{S}^{3}\right\}=TK{T}^{-1}$ $4$ ${S}_{4}{f}_{3478}=F$ $6$
${\left({S}_{4}\right)}_{{f}_{1458}}=\left\{1,T,{T}^{2},{T}^{3}\right\}=\left(S{T}^{3}\right)K{\left(S{T}^{3}\right)}^{-1}$ $4$ ${S}_{4}{f}_{1458}=F$ $6$
${\left({S}_{4}\right)}_{{f}_{2367}}=\left\{1,T,{T}^{2},{T}^{3}\right\}=\left(S{T}^{3}\right)K{\left(S{T}^{3}\right)}^{-1}$ $4$ ${S}_{4}{f}_{2367}=F$ $6$
${\left({S}_{4}\right)}_{{r}_{12345678}}={S}_{4}$ $24$ ${S}_{4}{r}_{12345678}=F$ $1$

 $1$ $v1$ $v2$ $v3$ $v4$ $v5$ $v6$ $v7$ $v8$ $S$ $v4$ $v1$ $v2$ $v3$ $v8$ $v5$ $v6$ $v7$ $S2$ $v3$ $v4$ $v1$ $v2$ $v7$ $v8$ $v5$ $v6$ $S3$ $v2$ $v3$ $v4$ $v1$ $v6$ $v7$ $v8$ $v5$ $T$ $v4$ $v3$ $v7$ $v8$ $v1$ $v2$ $v6$ $v5$ $ST$ $v8$ $v4$ $v3$ $v7$ $v5$ $v1$ $v2$ $v6$ $S2T$ $v7$ $v8$ $v4$ $v3$ $v6$ $v5$ $v1$ $v2$ $S3T$ $v3$ $v7$ $v8$ $v4$ $v2$ $v6$ $v5$ $v1$ $T2$ $v8$ $v7$ $v6$ $v5$ $v4$ $v3$ $v2$ $v1$ $ST2$ $v5$ $v8$ $v7$ $v6$ $v1$ $v4$ $v3$ $v2$ $S2T2$ $v6$ $v5$ $v8$ $v7$ $v2$ $v1$ $v4$ $v3$ $S3T2$ $v7$ $v6$ $v5$ $v8$ $v3$ $v2$ $v1$ $v4$ $T3$ $v5$ $v6$ $v2$ $v1$ $v8$ $v7$ $v3$ $v4$ $ST3$ $v1$ $v5$ $v6$ $v2$ $v4$ $v8$ $v7$ $v3$ $S2T3$ $v2$ $v1$ $v5$ $v6$ $v3$ $v4$ $v8$ $v7$ $S3T3$ $v6$ $v2$ $v1$ $v5$ $v7$ $v3$ $v4$ $v8$ $TS3$ $v1$ $v4$ $v8$ $v5$ $v2$ $v3$ $v7$ $v6$ $STS3$ $v5$ $v1$ $v4$ $v8$ $v6$ $v2$ $v3$ $v7$ $S2TS3$ $v8$ $v5$ $v1$ $v4$ $v7$ $v6$ $v2$ $v3$ $S3TS3$ $v4$ $v8$ $v5$ $v1$ $v3$ $v7$ $v6$ $v2$ $TS$ $v3$ $v2$ $v6$ $v7$ $v4$ $v1$ $v5$ $v8$ $STS$ $v7$ $v3$ $v2$ $v6$ $v8$ $v4$ $v1$ $v5$ $S2TS$ $v6$ $v7$ $v2$ $v6$ $v5$ $v8$ $v4$ $v1$ $S3TS$ $v2$ $v6$ $v7$ $v3$ $v1$ $v5$ $v8$ $v4$

## 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)