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)

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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