 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2010/b/6/2/solution

⟨ χ, χ ⟩ = 1

. Under the proposed vanishing condition, only the identity contributes, so

1 = \frac{χ ( 1 ) ^{2}}{∣ S _{6} ∣} = \frac{χ ( 1 ) ^{2}}{720} .

But

χ (1)

720

is not a square. Thus no such partition exists.

 New to topics? Read the docs here!