\(\mathbf{S}_4\)
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
S_{4} | e | C_{2} | C_{2}C_{2} | C_{3} | C_{4} | notes |
# | 1 | 6 | 3 | 8 | 6 | |S_{3}| = 24 |
A_{0} | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | -1 | 1 | 1 | -1 | alternating rep |
E | 2 | 0 | 2 | -1 | 0 | V_{4} acts trivially |
T_{1} | 3 | 1 | -1 | 0 | -1 | natural rep of tetrahedral group |
T_{2} | 3 | -1 | -1 | 0 | 1 | natural rep of octahedral group |
- All reps are absolutely irreducible (over Z)
- T_{1} is the representation corresponding to the group of all symmetries of the tetrahedron, denoted T_{d}, while T_{2} is the representation corresponding to the group O of rotational symmetries of the cube.
Representation ring
⊗ | A_{0} | A_{1} | E | T_{1} | T_{2} |
A_{0} | A_{0} | A_{1} | E | T_{1} | T_{2} |
A_{1} | A_{1} | A_{0} | E | T_{2} | T_{1} |
E | E | E | A_{0} + A_{1}+ E | T_{1}+T_{2} | T_{1}+T_{2} |
T_{1} | T_{1} | T_{2} | T_{1}+T_{2} | A_{0} + E + T_{1} + T_{2} | A_{1} + E + T_{1} + T_{2} |
T_{2} | T_{2} | T_{1} | T_{1}+T_{2} | A_{1} + E + T_{1} + T_{2} | A_{0} + E + T_{1} + T_{2} |
Permutation representations
Acting on the tetrahedron (T_{2})
- The permutation representation on 4 points (vertices of a tetrahedron) is A_{0} + T_{2}
- The permutation representation on the set of 6 edges of the tetrahedron is A_{0} + E + T_{2}
- The "orientation permutation" representation on the set of 4 faces of the tetrahedron is A_{1} + T_{1}
- The "orientation permutation" representation on the set of 6 edges of the tetrahedron is T_{1} + T_{2}
Acting on the cube (T_{1})
- The permutation on the 8 vertices of the cube is A_{0} + A_{1} + T_{1} + T_{2}
- The permutation representation on the 12 edges of the cube is A_{0} + E + T_{1} + 2T_{2}
- The permutation representation on the 6 faces of the cube is A_{0} + E + T_{1}
- The orientation permutation representation on the 24 oriented edges of the cube is the regular representation A_{0} + A_{1} + 2E + 3T_{1} + 3T_{2}
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 Z_{2}xZ_{2} is the subgroup generated by (1 2) and (3 4). On the other hand, V_{4} is the Klein Vierergruppe, consisting of the identity and the 3 elements conjugate to (1 2)(3 4): it is a normal subgroup of S_{4}.
Geometric realizations of orbit types:
- O_{1} := S_{4}/S_{4}: the tetrahedron or cube itself
- O_{2} := S_{4}/A_{4}: orientations of tetrahedron
- O_{3} := S_{4}/D_{4}: lines joining midpoints of opposite faces in cube
- O_{4} := S_{4}/S_{3}: vertices of tetrahedron
- O_{5} := S_{4}/V_{4}: ??
- O_{6} := S_{4}/Z_{4}: faces of cube
- O_{7} := S_{4}/Z_{3}: vertices of cube
- O_{8} := S_{4}/Z_{2}xZ_{2}: edges of tetrahedron
- O_{9} := S_{4}/Z_{2}: diagonals across faces of cube
- O_{10} := S_{4}/Z_{2}: oriented edges of tetrahedon; edges of cube
- O_{11} := S_{4}: generic point of tetrahedron or cube
Note that the actions on the tetrahedron and cube do not correspond to the same action on R^{3} (the former is the representation T_{1} while the latter is T_{2}). For example the element (1 2) ∈ S_{4} 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(Z_{3}, O_{4}) is the number of vertices of the tetrahedron fixed by the (a) subgroup Z_{3}, which is equal to 1.
1 | Z_{2} | Z_{2}' | Z_{2}xZ_{2}' | Z_{3} | Z_{4} | V_{4} | S_{3} | D_{4} | A_{4} | S_{4} | |
O_{11} | 24 | - | - | - | - | - | - | - | - | - | - |
O_{10} | 12 | 2 | - | - | - | - | - | - | - | - | - |
O_{9} | 12 | 0 | 4 | - | - | - | - | - | - | - | - |
O_{8} | 6 | 2 | 2 | 2 | - | - | - | - | - | - | - |
O_{7} | 8 | 0 | 0 | 0 | 2 | - | - | - | - | - | - |
O_{6} | 6 | 0 | 2 | 0 | 0 | 2 | - | - | - | - | - |
O_{5} | 6 | 0 | 6 | 0 | 0 | 0 | 6 | - | - | - | - |
O_{4} | 4 | 2 | 0 | 0 | 1 | 0 | 0 | 1 | - | - | - |
O_{3} | 3 | 1 | 3 | 1 | 0 | 1 | 3 | 0 | 1 | - | - |
O_{2} | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | - |
O_{1} | 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 | O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | O_{6} | O_{7} | O_{8} | O_{9} | O_{10} | O_{11} |
O_{1} | O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | O_{6} | O_{7} | O_{8} | O_{9} | O_{10} | O_{11} |
O_{2} | O_{2} | 2O_{2} | O_{5} | O_{7} | 2O_{5} | O_{9} | 2O_{7} | O_{9} | 2O_{9} | O_{11} | 2O_{11} |
O_{3} | O_{3} | O_{5} | O_{3}+O_{5} | O_{10} | 3O_{5} | O_{6}+O_{9} | O_{11} | O_{8}+O_{9} | 3O_{9} | O_{10}+O_{11} | 3O_{11} |
O_{4} | O_{4} | O_{7} | O_{10} | O_{4}+O_{10} | O_{11} | O_{11} | O_{7}+O_{11} | 2O_{10} | 2O_{11} | 2O_{10}+O_{11} | 4O_{11} |
O_{5} | O_{5} | 2O_{5} | 3O_{5} | O_{11} | 6O_{5} | 3O_{9} | 2O_{11} | 3O_{9} | 6O_{9} | 3O_{11} | 6O_{11} |
O_{6} | O_{6} | O_{9} | O_{6}+O_{9} | O_{11} | 3O_{9} | 2O_{6}+O_{11} | 2O_{11} | O_{9}+O_{11} | O_{9}+2O_{11} | 3O_{11} | 6O_{11} |
O_{7} | O_{7} | 2O_{7} | O_{11} | O_{7}+O_{11} | 2O_{11} | 2O_{11} | 2O_{7}+2O_{11} | 2O_{11} | 4O_{11} | 4O_{11} | 8O_{11} |
O_{8} | O_{8} | O_{9} | O_{8}+O_{9} | 2O_{10} | 3O_{9} | O_{9}+O_{11} | 2O_{11} | 2O_{8}+O_{11} | 2O_{9}+2O_{11} | 2O_{10}+2O_{11} | 6O_{11} |
O_{9} | O_{9} | 2O_{9} | 3O_{9} | 2O_{11} | 6O_{9} | O_{9}+2O_{11} | 4O_{11} | 2O_{9}+2O_{11} | 4O_{9}+4O_{11} | 6O_{11} | 12O_{11} |
O_{10} | O_{10} | O_{11} | O_{10}+O_{11} | 2O_{10}+O_{11} | 4O_{11} | 3O_{11} | 4O_{11} | 2O_{10}+2O_{11} | 6O_{11} | 2O_{10}+5O_{11} | 12O_{11} |
O_{11} | O_{11} | 2O_{11} | 3O_{11} | 4O_{11} | 6O_{11} | 6O_{11} | 8O_{11} | 6O_{11} | 12O_{11} | 12O_{11} | 24O_{11} |
Homomorphism β: Ω(G) → R(G)
- β(O_{1}) = A_{0}
- β(O_{2}) = A_{0} + A_{1}
- β(O_{3}) = A_{0} + E
- β(O_{4}) = A_{0} + T_{2}
- β(O_{5}) = A_{0} + A_{1} + 2E
- β(O_{6}) = A_{0} + E + T_{2}
- β(O_{7}) = A_{0} + A_{1} + T_{1} + T_{2}
- β(O_{8}) = A_{0} + E + T_{1}
- β(O_{9}) = A_{0} + A_{1} + 2E + T_{1} + T_{2}
- β(O_{10}) = A_{0} + E + 2T_{1} + T_{2}
- β(O_{11}) = A_{0} + A_{1} + 2E + 3T_{1} + 3T_{2}
Note that this is not injective.