# Symmetric Groups

The symmetric group Sn is of order n!. In these tables, elements are denoted as products of disjoint cycles. For example,

• C2 = (1 2)
• C2C2 = (1 2)(3 4)
• C3 = (1 2 3)
• C2C3 = (1 2)(3 4 5)

### S3

 S3 e C2 C3 notes # 1 3 2 |S3| = 6 A0 1 1 1 trivial rep A1 1 -1 1 alternating rep E 2 0 -1
• All reps are absolutely irreducible
• The permutation representation on 3 points (vertices of an equilateral triangle) is A0 + E
• The "orientation permutation" representation on the set of 3 edges of the triangle is A1 + E
• The "orientation" representation on the face of the equilateral triangle is A1
• Considering this triangle as a simplicial complex with an action of $$G=S_3$$, the boundary map is equivariant ('intertwining'), so we get

$0 \longrightarrow A_1 \stackrel{\partial_2}{\longrightarrow} A_1 \oplus E \stackrel{\partial_1}{\longrightarrow} A_0+E \longrightarrow 0.$

It follows that $$\partial_2$$ maps $$A_1$$ to $$A_1$$ (isomorphically) and $$\partial_1$$ maps $$A_1$$ to 0 and $$E$$ to $$E$$.

### S4

 S4 e C2 C2C2 C3 C4 notes # 1 6 3 8 6 |S3| = 24 A0 1 1 1 1 1 trivial rep A1 1 -1 1 1 -1 alternating rep E 2 0 2 -1 0 V4 acts trivially T1 3 1 -1 0 -1 natural rep of tetrahedral group T2 3 -1 -1 0 1 natural rep of octahedral group
• All reps are absolutely irreducible
• T1 is the representation corresponding to the group of all symmetries of the tetrahedron, denoted Td, while T2 is the representation corresponding to the group O of rotational symmetries of the cube.
• The permutation representation on 4 points (vertices of a tetrahedron) is A0 + T1
• The permutation representation on the set of 6 edges of the tetrahedron is A0 + E + T1
• The "orientation permutation" representation on the set of 6 edges of the tetrahedron is T1 + T2
• The "orientation permutation" representation on the set of 4 faces of the tetrahedron is A1 + T2
• When acting on the cube, the permutation on the 8 vertices is A0 + A1 + T1 + T2
• The permutation representation on the 12 edges of the cube is A0 + E + 2T1 + T2
• The permutation representation on the 6 faces of the cube is A0 + E + T2
• The simplicial complex (tetrahedron) with action of $$S_4$$ is

$0 \longrightarrow A_1 \stackrel{\partial_3}{\longrightarrow} A_1\oplus T_2 \stackrel{\partial_2}{\longrightarrow} T_1 \oplus T_2 \stackrel{\partial_1}{\longrightarrow} A_0+T_1 \longrightarrow 0.$

### S5

 S5 e C2 C2C2 C3 C2C3 C4 C5 notes # 1 10 15 20 20 30 24 |S5| = 120 A0 1 1 1 1 1 1 1 trivial rep A1 1 -1 1 1 -1 -1 1 alternating rep G1 4 2 0 1 -1 0 -1 G2 4 -2 0 1 1 0 -1 = G1 ⊗ A1 H1 5 1 1 -1 1 -1 0 H2 5 -1 1 -1 -1 1 0 = H1 ⊗ A1 J 6 0 -2 0 0 0 1
• The names for the reps are not standard (are there any standard names?)
• Permutation rep on 5 points (vertices of simplex in R4) is A0 ⊕ G1
• Permutation rep on the 10 edges of the simplex is A0 ⊕ G1 ⊕ H1
• Orientation rep on the the 10 edges is G1 ⊕ J

### S6

 S6 e C2 C2C2 C2C2C2 C3 C2C3 C3C3 C4 C2C4 C5 C6 notes # 1 15 45 15 40 120 40 90 90 144 120 |S6| = 720 A0 1 1 1 1 1 1 1 1 1 1 1 trivial rep A1 1 -1 1 -1 1 -1 1 -1 1 1 -1 alternating rep H1 5 3 1 -1 2 0 -1 1 -1 0 -1 H2 5 -3 1 1 2 0 -1 -1 -1 0 1 = H1 ⊗ A1 H3 5 1 1 -3 -1 1 2 -1 -1 0 0 H4 5 -1 1 3 -1 -1 2 1 -1 0 0 = H3 ⊗ A1 M1 9 3 1 3 0 0 0 -1 1 -1 0 M2 9 -3 1 -3 0 0 0 1 1 -1 0 = M1 ⊗ A1 N1 10 2 -2 -2 1 1 1 0 0 0 -1 N2 10 -2 -2 2 1 1 1 0 0 0 -1 = N1 ⊗ A1 U 16 0 0 0 -2 0 -2 0 0 1 0
• The names for the reps are not standard
• Permutation rep on 6 points (vertices of simplex in R5) is A0 ⊕ H1
• Permutation rep on the 15 edges of the simplex is A0 ⊕ H1 ⊕ M1
• Permutation rep on the 20 faces of the simplex is A0 ⊕ H1 ⊕ H3 ⊕ M1