Cubic Groups
The cubic groups are the groups of symmetries of the platonic solids, T, T_{d}, O, O_{h} , I and I_{h}.
On this page:
Cases treated elsewhere:
- T ≅ A_{4} (T is the group of rotations of the tetrahedron)
- T_{d} ≅ O ≅ S_{4} (T_{d} is the group of all symmetries of the tetrahedron)
- I ≅ A_{5} (I is the group of rotations of the icosahedron)
The final column marked "Mull" is the Mulliken symbol used in Physics and Chemistry.
T_{h} is the group of symmetries of a `decorated cube'
The 'decorated cube' is as follows. Take a cube and draw a single new line across each face that divides the square face into two equal rectangles - and this should be done in such a way that each edge of the cube only meets one of these lines (so each edge meets exactly 3 of the rectangles).
T_{h} has order 24 and is isomorphic to \(\mathbf{A}_4\times\mathbb{Z}_2^c\), where \(\mathbb{Z}_2^c\) is the centre of O(3), generated by `inversion' \(i=-I:\mathbf{x}\mapsto-\mathbf{x}\), and often denoted C_{i}. The representations are therefore obtained from those of A_{4} tensored with those of \(\mathbb{Z}_2\).
T_{h} | e | C_{2}C_{2} | C_{3} | C_{3}^{2} | i | iC_{2}C_{2} | iC_{3} | iC_{3}^{2} | notes |
# | 1 | 3 | 4 | 4 | 1 | 3 | 4 | 4 | |T_{h}|=24 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | alternating rep |
E_{1} | 2 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | |
E_{2} | 2 | 2 | -1 | -1 | -2 | -2 | 1 | 1 | A_{1} ⊗ E_{1} |
T_{1} | 3 | -1 | 0 | 0 | 3 | -1 | 0 | 0 | A_{1} ⊗ T_{2} |
T_{2} | 3 | -1 | 0 | 0 | -3 | 1 | 0 | 0 | natural rep on cube |
- T_{h} is the symmetry group of a volleyball with its markings, and T_{2} is the representation in question.
- T_{1} is the representation where i acts trivially, so factors through the rotation group T.
O_{h} is the group of all symmetries of the cube
O_{h} has order 48 and is isomorphic to \(\mathbf{S}_4\times\mathbb{Z}_2^c\), where \(\mathbb{Z}_2^c\) is the centre of O(3), generated by `inversion' \(i=-I:\mathbf{x}\mapsto-\mathbf{x}\), and often denoted C_{i}
Notation for elements:
- C_{k} is a rotation of order k (C_{4} is a rotation by π/2; C_{2} is a rotation by π around a line through mid points of opposite edges)
- iC_{k} is C_{k} composed with i.
O_{h} | e | C_{4} | C_{4}^{2} | C_{3} | C_{2} | i | iC_{4} | iC_{4}^{2} | iC_{3} | iC_{2} | notes | Mull. |
# | 1 | 6 | 3 | 8 | 6 | 1 | 6 | 3 | 8 | 6 | |O_{h}| = 48 | |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep | A_{1g} |
A_{1} | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | alternating rep | A_{1u} |
A_{2} | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | = A_{1} ⊗ A_{3} | A_{2g} |
A_{3} | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | = A_{1} ⊗ A_{2} | A_{2u} |
E_{1} | 2 | 0 | 2 | -1 | 0 | 2 | 0 | 2 | -1 | 0 | E_{g} | |
E_{2} | 2 | 0 | 2 | -1 | 0 | -2 | 0 | -2 | 1 | 0 | = E_{1} ⊗ A_{1} | E_{u} |
T_{1} | 3 | 1 | -1 | 0 | -1 | -3 | -1 | 1 | 0 | 1 | symmetry rep of cube | T_{1u} |
T_{2} | 3 | 1 | -1 | 0 | -1 | 3 | 1 | -1 | 0 | -1 | = T_{1} ⊗ A_{1} | T_{1g} |
T_{3} | 3 | -1 | -1 | 0 | 1 | -3 | 1 | 1 | 0 | -1 | = T_{1} ⊗ A_{2} | T_{2u} |
T_{4} | 3 | -1 | -1 | 0 | 1 | 3 | -1 | -1 | 0 | 1 | = T_{1} ⊗ A_{3} | T_{2g} |
- All reps are absolutely irreducible — even over Q
- The permutation representation on the set of 8 vertices of the cube is A_{0} + A_{3} + T_{1} + T_{4} = (A_{0} + A_{3})⊗(A_{0} + T_{1})
- The permutation representation on the set of 6 vertices of the octahedron is A_{0} + E_{1} + T_{1}
- The permutation representation on the set of 12 edges of either is A_{0} + E_{2} + T_{1} + T_{3} + T_{4}
- The permutation representation on the set of 3 diagonals of the octahedron is A_{0} + E_{1} _{}
- The permutation representation on the set of 4 diagonals of the cube is A_{0} + T_{4} _{}
- The "orientation permutation" representation on the set of 4 oriented diagonals of the cube is A_{3} + T_{1}
- The "orientation permutation" representation on the set of 6 faces of the cube is A_{1} +??_{}
- The "orientation permutation" representation on the set of 8 faces of the octahedron is T_{1} + ??
- The cube contains 2 inscribed tetrahedra; the permutation rep of this set is A_{0} + A_{3}
I_{h} is the group of all symmetries of the icosahedron
It is isomorphic to A_{5} x C_{i}.
I_{h} | e | C_{5} | C_{5}^{2} | C_{3} | C_{2} | i | iC_{5} | iC_{5}^{2} | iC_{3} | iC_{2} | notes | Mull. |
# | 1 | 12 | 12 | 20 | 15 | 1 | 12 | 12 | 20 | 15 | |I_{h}|=120 | - |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep | A_{1g} |
A_{1} | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | alternating rep | A_{1u} |
T_{1} | 3 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | -3 | -\(\varphi^+\) | -\(\varphi^-\) | 0 | 1 | Symetry of icosahedron | T_{1u} |
T_{2} | 3 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | 3 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | = T_{1} ⊗ A_{1} | T_{1g} |
T_{3} | 3 | \(\varphi^-\) | \(\varphi^+\) | 0 | -1 | -3 | -\(\varphi^-\) | -\(\varphi^+\) | 0 | 1 | T_{2u} | |
T_{4} | 3 | \(\varphi^-\) | \(\varphi^+\) | 0 | -1 | 3 | \(\varphi^-\) | \(\varphi^+\) | 0 | -1 | = T_{3} ⊗ A_{1} | T_{2g} |
G_{1} | 4 | -1 | -1 | 1 | 0 | -4 | 1 | 1 | -1 | 0 | G_{u} | |
G_{2} | 4 | -1 | -1 | 1 | 0 | 4 | -1 | -1 | 1 | 0 | = G_{1} ⊗ A_{1} | G_{g} |
H_{1} | 5 | 0 | 0 | -1 | 1 | -5 | 0 | 0 | 1 | -1 | H_{u} | |
H_{2} | 5 | 0 | 0 | -1 | 1 | 5 | 0 | 0 | -1 | 1 | = H_{1} ⊗ A_{1} | H_{g} |
- \(\varphi^+\) = 2cos(π/5) = ½(1+√5) (=golden ratio), and \(\varphi^-\) = -2cos(2π/5) = ½(1-√5) ( = -(\(\varphi^+\))^{-1})
- All reps are absolutely irreducible.
- As a subgroup of O(3), I_{h} contains the central element i:x → -x (inversion in the origin).
So showing that I_{h} = I x C_{i} = A_{5} x C_{i}.