Alternating Groups
The alternating group An is of order (½)n!. In these tables, elements are denoted as products of disjoint cycles. For example,
- C2C2 = (1 2)(3 4)
- C3 = (1 2 3)
On this page:
- A4 ≅ T, the group of rotational symmetries of the tetrahedron
- A5 ≅ I, the group of rotational symmetries of the icosahedron
- A6
A4
A4
| e
| C2C2
| C3
| C32
| notes
|
#
| 1
| 3
| 4
| 4
| |A4|=12
|
A0
| 1
| 1
| 1
| 1
| trivial rep
|
E
| 2
| 2
| -1
| -1
| not absolutely irreducible
|
T
| 3
| -1
| 0
| 0
| natural rep of tetrahedral group
|
- The natural permutation representation on 4 objects is \(A_0 + T\)
- The permutation representation on the 6 edges of the tetrahedron is \(A_0 + E + T\)
- The permutation representation on the 3 'diagonals' (joining mid-points of opposite edges) is \(A_0+E\). Each of these diagonals is fixed by the Klein 4-group \(V_4\).
- The "oriented permutation" representation on the 6 edges of the tetrahedron is \(2T\)
- \(E\) is of complex type, and its complexification splits over \(\mathbf{C}\) as (1, 1, ω, ω²) ⊕ (1, 1, ω², ω), where \(\omega=\frac12(-1+i\sqrt{3})\) is a cube root of unity.
A5
A5
| e
| C2C2
| C3
| C5
| C52
| notes
|
#
| 1
| 15
| 20
| 12
| 12
| |A5|=60
|
A0
| 1
| 1
| 1
| 1
| 1
| trivial rep
|
T1
| 3
| -1
| 0
| \(\varphi^+\)
| \(\varphi^-\)
| Symmetries of icosahedron
|
T2
| 3
| -1
| 0
| \(\varphi^-\)
| \(\varphi^+\)
|
|
G
| 4
| 0
| 1
| -1
| -1
|
|
H
| 5
| 1
| -1
| 0
| 0
|
|
- \(\varphi^+ = 2\cos(\pi/5) = \frac12(1+\sqrt5)\) (=golden ratio) and
\(\varphi^- = -2\cos(2\pi/5) = \frac12(1-\sqrt5)\) ( = \(-(\varphi^+)^{-1}\))
- All the representations are absolutely irreducible
- \(T_1\) and \(T_2\) are related by an outer automorphism of the group.
- The representation \(T := T_1 + T_2\) is irreducible over \(\mathbb{Q}\), but not (of course) absolutely irreducible; it becomes reducible over \(\mathbb{Q}[\sqrt5]\). It is also the restriction to \(A_5\) of the 6-dimensional irreducible representation of \(S_5\), see here.
- \(A_5\) is the group of rotational symmetries of the regular icosahedron (and dodecahedron), so denoted I in Schoenflies. If C5 = (1 2 3 4 5) acts by rotations by \(2\pi/5\) then this geometric representation is T1.
- The natural permutation representation on 5 objects is \(A_0 + G\)
- The permutation representation on the set of 12 vertices of the icosahedron is \(A_0 + T + H\)
- The permutation representation on the set of 20 vertices of the dodecahedron is \(A_0 + T + 2G + H\)
- The permutation representation on the set of 30 edges of either is \(A_0 + T + 2G + 3H\)
- The "oriented permutation" representation on the set of 30 edges of either is \(2(T + G + H)\)
- The permutation representation on the set of 6 diagonals of the icosahedron is \(A_0 + H\)
- The dodecahedron famously contains 5 inscribed tetrahedra (each formed by joining 4 non-adjacent vertices) (also 5 cubes). The resulting permutation representation is \(A_0 + G\).
A6
A6
| e
| C2C2
| C2C4
| C3
| C3C3
| C5
| C52
| notes
|
#
| 1
| 45
| 90
| 40
| 40
| 72
| 72
| |A6|=360
|
A0
| 1
| 1
| 1
| 1
| 1
| 1
| 1
| trivial rep
|
H1
| 5
| 1
| -1
| 2
| -1
| 0
| 0
|
|
H2
| 5
| 1
| -1
| -1
| 2
| 0
| 0
|
|
L1
| 8
| 0
| 0
| -1
| -1
| \(\varphi^+\)
| \(\varphi^-\)
|
|
L2
| 8
| 0
| 0
| -1
| -1
| \(\varphi^-\)
| \(\varphi^+\)
|
|
M
| 9
| 1
| 1
| 0
| 0
| -1
| -1
|
|
N
| 10
| -2
| 0
| 1
| 1
| 0
| 0
|
|
- Names of reps are not standard
- \(\varphi^+\) and \(\varphi^-\) are as for A5 above
- L1 + L2 is the restriction to A6 of the representation U of S6.