1. Why is the permutation representation (of Sn) not irreducible? Hint: You may start by looking-at the generators of Sn.
2. Using the result above and its decomposition into irreducible representations show that χs4 (ρperm (1, 2, 3)) = 1. Hint: this exercise does involve more than one idea and is not the easiest one on this sheet.
3. What is the element of largest order in S12?
4. Show that the character table on the left below can be completed to the form given on the right. Give a brief and concise explanation of each step.
x ≡ (-1 + i√3)/2: y = (-1 + i√7)/2.
|
C1
|
C2
|
C3
|
C4
|
C5
|
|
|Ci|
|
1
|
|
3
|
7
|
7
|
|
χ(ρ2)
|
1
|
|
1
|
x
|
x*
|
|
χ(ρ4)
|
|
y
|
|
0
|
0
|
|
C1
|
C2
|
C3
|
C4
|
C5
|
|
|Ci|
|
1
|
3
|
3
|
7
|
7
|
|
χ(ρ1)
|
1
|
1
|
1
|
1
|
1
|
|
χ(ρ2)
|
1
|
1
|
1
|
x
|
x*
|
|
χ(ρ3)
|
1
|
1
|
1
|
x*
|
x*
|
|
χ(ρ4)
|
3
|
y
|
y*
|
0
|
0
|
|
χ(ρ5)
|
3
|
y*
|
y*
|
0
|
0
|