 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2007/b/5/c/solution

μ = (m, ν)

χ^{λ} (m, ν) = \sum_{λ / ρ a rim hook of size m} (- 1)^{ht (λ / ρ)} χ^{ρ} (ν) .

The sum is over partitions

ρ

whose diagrams remain after removing the specified border strip;

ν

is the remaining cycle type, and the recursion ends with the empty character equal to one. At the identity,

χ^{ρ} (1^{∣ ρ ∣}) = f^{ρ}

. Incompatible total sizes contribute zero.

(2, 2)

and a transposition, a horizontal strip in the bottom row leaves

(2)

with positive sign; a vertical strip in the right column leaves

(1, 1)

with negative sign. Both remaining dimensions are one. Thus

χ^{(2, 2)} (2, 1, 1) = 1 - 1 = 0.

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