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 A0 + T
- The permutation representation on the 6 edges of the tetrahedron is A0 + E + T
- The "oriented permutation" representation on the 6 edges of the tetrahedron is 2T
- E is of complex type, and its complexification splits over C as (1, 1, ω, ω²) ⊕ (1, 1, ω², ω), where ω = ½(-1+√3 i) 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
| \(\gamma^+\)
| \(\gamma^-\)
| Symmetries of icosahedron
|
| T2
| 3
| -1
| 0
| \(\gamma^-\)
| \(\gamma^+\)
|
|
| G
| 4
| 0
| 1
| -1
| -1
|
|
| H
| 5
| 1
| -1
| 0
| 0
|
|
- \(\gamma^+ = -2\cos(2\pi/5) = \frac12(1+\sqrt5)\) (=golden ratio) and
\(\gamma^- = -2\cos(\pi/5) = \frac12(1-\sqrt5)\) ( = \(-(\gamma^+)^{-1}\))
- All the representations are absolutely irreducible
- T1 and T 2 are related by an outer automorphism of the group.
- The representation T := T1 + T2 is irreducible over \(\mathbb{Q}\), but not (of course) absolutely irreducible; it becomes reducible over \(\mathbb{Q}[\sqrt5]\).
- A5 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 A0 + G
- The permutation representation on the set of 12 vertices of the icosahedron is A0 + T + H
- The permutation representation on the set of 20 vertices of the dodecahedron is A0 + T + 2G + H
- The permutation representation on the set of 30 edges of either is A0 + 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 A0 + H
- The dodecahedron famously contains 5 inscribed tetrahedra (each formed by joining 4 non-adjacent vertices) (also 5 cubes). The resulting permutation representation is A0 + 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
| \(\gamma^+\)
| \(\gamma^-\)
|
|
| L2
| 8
| 0
| 0
| -1
| -1
| \(\gamma^-\)
| \(\gamma^+\)
|
|
| M
| 9
| 1
| 1
| 0
| 0
| -1
| -1
|
|
| N
| 10
| -2
| 0
| 1
| 1
| 0
| 0
|
|
- Names of reps are not standard
- \(\gamma^+\) and \(\gamma^-\) are as for A5 above
- L1 + L2 is the restriction to A6 of the representation U of S6.