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

The group action of A4 as rotations of a tetrahedron

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 10 February 2011

The group action of A4 as rotations of a tetrahedron

A4 is the group of rotations of the tetrahedron. We shall denote the vertices by vi, the edge connecting the vertex i to the vertex j by eij, i<j, and the face adjacent to the three vertices vi, vj, vk, by fijk, i<j<k. Let r1234 denote the region determined by the the inside of the tetrahedron. Let pij, 1i,j4 denote the point on the edge connecting vi to vj which is a third of the way from vi to vj.

T v3 v2 v1 v4 S

Let S be the 60° rotation about the bottom face taking v1 v2 v3 v1 and fixing v4. Let T be the 180° rotation about the line connecting the midpoint of the edge e34 with the midpoint of the edge e12, taking v1 v2and v3v4. Note that S3=1, T2=1, and ST3=1.

Let P= pij 1i,j4 , 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 A4 acts on the tetrahedron, A4 acts on each of these sets.

Stabilizer Size of Stabilizer Orbit Size of Orbit
A4 pij= 1 1 A4pij= P 12
A4v4= 1,S,S2 =H 3 A4v4= V 4
A4v3= 1,TST-1, TS2T-1 = THT-1 3 A4v3=V 4
A4v1= 1,TS,S2T = STHST-1 3 A4v1=V 4
A4v2= 1,ST, ST2 = S2T HS2T-1 3 A4v3=V 4
A4e12= 1, T 2 A4 e12=E 6
A4e34= 1, T 2 A4 e34=E 6
A4e14= 1, STS-1 2 A4 e14=E 6
A4e23= 1, STS-1 2 A4 e23=E 6
A4e13= 1, S2TS-2 2 A4 e13=E 6
A4e24= 1, S2TS-2 2 A4 e24=E 6
A4 f123 = 1, S, S2 3 A4 f123 =F 4
A4 f124 = 1, TST-1, TS2T-1 3 A4 f124 =F 4
A4 f234 = 1, ST S ST-1, ST S2 ST-1 3 A4 f234 =F 4
A4 f134 = 1, S2T S S2T-1, S2T S2 S2T-1 3 A4 f134 =F 4
A4 r1234 = A4 12 A4 r1234 =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


[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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