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

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Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 5 / d / Solution by

Codex  0 2026-10-31

For shape

(n - 2, 2)

with

n \geq 6

, there are exactly two removable rim hooks of size two: the bottom row, leaving

(n - 2)

, and the final two cells of the first row, leaving

(n - 4, 2)

. Both have height zero. A vertical pair cannot be removed while leaving a Young diagram because the first row is longer than the second by at least two.

By the Murnaghan–Nakayama rule and part (b),

χ^{(n - 2, 2)} (2, 1^{n - 2}) = 1 + f^{(n - 4, 2)} = 1 + (2 n - 2) - (1 n - 2) = \frac{( n - 4 ) ( n - 3 )}{2} .

(1)

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