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Further details

• S₃ S₄

\(\mathbf{S}_4\)

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

• All reps are absolutely irreducible (over Z)
• T₁ is the representation corresponding to the group of all symmetries of the tetrahedron, denoted T_d, while T₂ is the representation corresponding to the group O of rotational symmetries of the cube.

Representation ring

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: \(\mathbb{Z}_2\) refers to the group generated by (1 2) (or any conjugate subgroup), while \(\mathbb{Z}_2\) (?) 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 Klein Vierergruppe, consisting of the identity and the 3 elements conjugate to (1 2)(3 4): 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 O_r = 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.

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).

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.