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:

  • A4T, the group of rotational symmetries of the tetrahedron
  • A5I, 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.