Symmetric Groups
The symmetric group S_{n} is of order n!. In these tables, elements are denoted as products of disjoint cycles. For example,
- C_{2} = (1 2)
- C_{2}C_{2} = (1 2)(3 4)
- C_{3} = (1 2 3)
- C_{2}C_{3} = (1 2)(3 4 5)
On this page:
S_{3}
S_{3} | e | C_{2} | C_{3} | notes |
# | 1 | 3 | 2 | |S_{3}| = 6 |
A_{0} | 1 | 1 | 1 | trivial rep |
A_{1} | 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 A_{0} + E
- The "orientation permutation" representation on the set of 3 edges of the triangle is A_{1} + E
- The "orientation" representation on the face of the equilateral triangle is A_{1}
- 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 \(E\).
S_{4}
S_{4} | e | C_{2} | C_{2}C_{2} | C_{3} | C_{4} | notes |
# | 1 | 6 | 3 | 8 | 6 | |S_{3}| = 24 |
A_{0} | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | -1 | 1 | 1 | -1 | alternating rep |
E | 2 | 0 | 2 | -1 | 0 | V_{4} acts trivially |
T_{1} | 3 | 1 | -1 | 0 | -1 | natural rep of tetrahedral group |
T_{2} | 3 | -1 | -1 | 0 | 1 | natural rep of octahedral group |
- All reps are absolutely irreducible
- T_{1} is the representation corresponding to the group of all symmetries of the tetrahedron, denoted T_{d}, while T_{2} 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 A_{0} + T_{1}
- The permutation representation on the set of 6 edges of the tetrahedron is A_{0} + E + T_{1}
- The "orientation permutation" representation on the set of 6 edges of the tetrahedron is T_{1} + T_{2}
- The "orientation permutation" representation on the set of 4 faces of the tetrahedron is A_{1} + T_{2}
- When acting on the cube, the permutation on the 8 vertices is A_{0} + A_{1} + T_{1} + T_{2}
- The permutation representation on the 12 edges of the cube is A_{0} + E + 2T_{1} + T_{2}
- The permutation representation on the 6 faces of the cube is A_{0} + E + T_{2}
- The simplicial complex (tetrahedron) with action of \(S_4\) is
\[0 \longrightarrow A_1 \stackrel{\partial_3}{\longrightarrow} A_1\oplus T_2 \stackrel{\partial_2}{\longrightarrow} T_1 \oplus T_2 \stackrel{\partial_1}{\longrightarrow} A_0+T_1 \longrightarrow 0.\]
S_{5}
S_{5} | e | C_{2} | C_{2}C_{2} | C_{3} | C_{2}C_{3} | C_{4} | C_{5} | notes |
# | 1 | 10 | 15 | 20 | 20 | 30 | 24 | |S_{5}| = 120 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | -1 | 1 | 1 | -1 | -1 | 1 | alternating rep |
G_{1} | 4 | 2 | 0 | 1 | -1 | 0 | -1 | |
G_{2} | 4 | -2 | 0 | 1 | 1 | 0 | -1 | = G_{1} ⊗ A_{1} |
H_{1} | 5 | 1 | 1 | -1 | 1 | -1 | 0 | |
H_{2} | 5 | -1 | 1 | -1 | -1 | 1 | 0 | = H_{1} ⊗ A_{1} |
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 R^{4}) is A_{0} ⊕ G_{1}
- Permutation rep on the 10 edges of the simplex is A_{0} ⊕ G_{1} ⊕ H_{1}
- Orientation rep on the the 10 edges is G_{1} ⊕ J
S_{6}
S_{6} | e | C_{2} | C_{2}C_{2} | C_{2}C_{2}C_{2} | C_{3} | C_{2}C_{3} | C_{3}C_{3} | C_{4} | C_{2}C_{4} | C_{5} | C_{6} | notes |
# | 1 | 15 | 45 | 15 | 40 | 120 | 40 | 90 | 90 | 144 | 120 | |S_{6}| = 720 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | alternating rep |
H_{1} | 5 | 3 | 1 | -1 | 2 | 0 | -1 | 1 | -1 | 0 | -1 | |
H_{2} | 5 | -3 | 1 | 1 | 2 | 0 | -1 | -1 | -1 | 0 | 1 | = H_{1} ⊗ A_{1} |
H_{3} | 5 | 1 | 1 | -3 | -1 | 1 | 2 | -1 | -1 | 0 | 0 | |
H_{4} | 5 | -1 | 1 | 3 | -1 | -1 | 2 | 1 | -1 | 0 | 0 | = H_{3} ⊗ A_{1} |
M_{1} | 9 | 3 | 1 | 3 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | |
M_{2} | 9 | -3 | 1 | -3 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | = M_{1} ⊗ A_{1} |
N_{1} | 10 | 2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | |
N_{2} | 10 | -2 | -2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | = N_{1} ⊗ A_{1} |
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 R^{5}) is A_{0} + H_{1}
- Permutation rep on the 15 edges of the simplex is A_{0} + H_{1} + M_{1}
- Permutation rep on the 20 faces of the simplex is A_{0} + H_{1} + H_{3} + M_{1}