\(\mathbf{S}_4\)

This is the symmetric group on four things. It can be realized as the group of symmetries of a tetrahedron (in its incarnation as \(\mathbb{T}_d\)), and as the group of rotational symmetries of the cube (where it is called \(\mathbb{O}\)).

Character table

S4 e C2 C2C2 C3 C4 notes
# 1 6 3 8 6 |S3| = 24
A0 1 1 1 1 1 trivial rep
A1 1 -1 1 1 -1 alternating rep
E 2 0 2 -1 0 V4 acts trivially
T1 3 1 -1 0 -1 natural rep of tetrahedral group
T2 3 -1 -1 0 1 natural rep of octahedral group
  • All reps are absolutely irreducible (over Z)
  • T1 is the representation corresponding to the group of all symmetries of the tetrahedron, denoted Td, while T2 is the representation corresponding to the group O of rotational symmetries of the cube.

Representation ring

A0 A1 E T1 T2
A0 A0 A1 E T1 T2
A1 A1 A0 E T2 T1
E E E A0 + A1+ E T1+T2 T1+T2
T1 T1 T2 T1+T2 A0 + E + T1 + T2 A1 + E + T1 + T2
T2 T2 T1 T1+T2 A1 + E + T1 + T2 A0 + E + T1 + T2

Permutation representations

Acting on the tetrahedron (T2)

  • The permutation representation on 4 points (vertices of a tetrahedron) is A0 + T2
  • The permutation representation on the set of 6 edges of the tetrahedron is A0 + E + T2
  • The "orientation permutation" representation on the set of 4 faces of the tetrahedron is A1 + T1
  • The "orientation permutation" representation on the set of 6 edges of the tetrahedron is T1 + T2

Acting on the cube (T1)

  • The permutation on the 8 vertices of the cube is A0 + A1 + T1 + T2
  • The permutation representation on the 12 edges of the cube is A0 + E + T1 + 2T2
  • The permutation representation on the 6 faces of the cube is A0 + E + T1
  • The orientation permutation representation on the 24 oriented edges of the cube is the regular representation A0 + A1 + 2E + 3T1 + 3T2

Burnside ring Ω(G)

Notation: Z2 refers to the group generated by (1 2) (or any conjugate subgroup), while Z2 refers to the subgroup generated by (1 2)(3 4). Then Z2xZ2 is the subgroup generated by (1 2) and (3 4). On the other hand, V4 is the Klien Vierergruppe, consisting of the 3 elements conjugate to (1 2)(3 4) and the identity: it is a normal subgroup of S4.

Geometric realizations of orbit types:

  • O1 := S4/S4: the tetrahedron or cube itself
  • O2 := S4/A4: orientations of tetrahedron
  • O3 := S4/D4: lines joining midpoints of opposite faces in cube
  • O4 := S4/S3: vertices of tetrahedron
  • O5 := S4/V4: ??
  • O6 := S4/Z4: faces of cube
  • O7 := S4/Z3: vertices of cube
  • O8 := S4/Z2xZ2: edges of tetrahedron
  • O9 := S4/Z2: diagonals across faces of cube
  • O10 := S4/Z2: oriented edges of tetrahedon; edges of cube
  • O11 := S4: generic point of tetrahedron or cube

Note that the actions on the tetrahedron and cube do not correspond to the same action on R3 (the former is the representation T1 while the latter is T2). For example the element (1 2) ∈ S4 acts on the tetrahedron by a reflexion exchanging two vertices, and it acts on the cube by a rotation by π about an axis joining mid-points of opposite edges.

Table of marks:

The rows are the orbit types, the columns are the subgroups. The entries represent #Fix(H, G/K) - the number of elements in the orbit Or = G/K fixed by the subgroup H.

For example, #Fix(Z3, O4) is the number of vertices of the tetrahedron fixed by the (a) subgroup Z3, which is equal to 1.

1 Z2 Z2' Z2xZ2' Z3 Z4 V4 S3 D4 A4 S4
O11 24 - - - - - - - - - -
O10 12 2 - - - - - - - - -
O9 12 0 4 - - - - - - - -
O8 6 2 2 2 - - - - - - -
O7 8 0 0 0 2 - - - - - -
O6 6 0 2 0 0 2 - - - - -
O5 6 0 6 0 0 0 6 - - - -
O4 4 2 0 0 1 0 0 1 - - -
O3 3 1 3 1 0 1 3 0 1 - -
O2 2 0 2 0 2 0 2 0 0 2 -
O1 1 1 1 1 1 1 1 1 1 1 1

Product structure in Burnside Ring Ω(G):

The Cartesian product of two orbit types decomposes as a union of disjoint orbits, and this defines the product structure in the Burnside ring. Since the rows in the table above are linearly independent, this product is easily deduced from that table (componentwise multiplication of the rows).

X O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11
O1 O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11
O2 O2 2O2 O5 O7 2O5 O9 2O7 O9 2O9 O11 2O11
O3 O3 O5 O3+O5 O10 3O5 O6+O9 O11 O8+O9 3O9 O10+O11 3O11
O4 O4 O7 O10 O4+O10 O11 O11 O7+O11 2O10 2O11 2O10+O11 4O11
O5 O5 2O5 3O5 O11 6O5 3O9 2O11 3O9 6O9 3O11 6O11
O6 O6 O9 O6+O9 O11 3O9 2O6+O11 2O11 O9+O11 O9+2O11 3O11 6O11
O7 O7 2O7 O11 O7+O11 2O11 2O11 2O7+2O11 2O11 4O11 4O11 8O11
O8 O8 O9 O8+O9 2O10 3O9 O9+O11 2O11 2O8+O11 2O9+2O11 2O10+2O11 6O11
O9 O9 2O9 3O9 2O11 6O9 O9+2O11 4O11 2O9+2O11 4O9+4O11 6O11 12O11
O10 O10 O11 O10+O11 2O10+O11 4O11 3O11 4O11 2O10+2O11 6O11 2O10+5O11 12O11
O11 O11 2O11 3O11 4O11 6O11 6O11 8O11 6O11 12O11 12O11 24O11

Homomorphism β: Ω(G) → R(G)

  • β(O1) = A0
  • β(O2) = A0 + A1
  • β(O3) = A0 + E
  • β(O4) = A0 + T2
  • β(O5) = A0 + A1 + 2E
  • β(O6) = A0 + E + T2
  • β(O7) = A0 + A1 + T1 + T2
  • β(O8) = A0 + E + T1
  • β(O9) = A0 + A1 + 2E + T1 + T2
  • β(O10) = A0 + E + 2T1 + T2
  • β(O11) = A0 + A1 + 2E + 3T1 + 3T2

Note that this is not injective.