 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2008/b/6/a/solution

If the cycle partition contains a part

r

χ^{λ} ((r, ν)) = μ \subseteq λ λ / μ a border strip of size r \sum (- 1)^{ht (λ / μ)} χ^{μ} (ν) .

2 \times 2

block; its height is its number of occupied rows minus one. The remaining boxes must be a partition diagram. The recursion starts with character one for the empty diagram at the empty cycle type, and an empty sum is zero. Repeated removal for all cycle lengths computes the character value.

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