# 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)

• 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.