Theory

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Character tables

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Further details

• S₃ S₄

Alternating Groups

The alternating group A_n 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

• 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

Same character table as complex irreducibles.

• \(\varphi^+ = 2\cos(\pi/5) = \frac12(1+\sqrt5)\) (=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 \(A_0 + G\).
• The group also acts on the 4-simplex (with 5 vertices in \(\mathbf{R}^5\)). The permutation representation on the set of 10 edges is \(A_0+G+H\) (and on the 5 vertices is \(A_0+G\) as above).

A₆

Same as table for complex irreducibles: these are all absolutely irreducible

• Names of reps are not standard!
• \(\varphi^+\) and \(\varphi^-\) are as for A₅ above
• \(A_0+H_1\) is standard rep on \(\mathbf{R}^6\), and equivalently permutation rep on 6 objects (eg vertices of 5-simplex)
• \(A_0+H_1+M\) is permutation rep on the set of 15 edges of the 5-simplex.
• L₁ + L₂ is the restriction to A₆ of the representation U of S₆.