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)
On this page:
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 (solid) 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