Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 2 / a / i

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- Solution i

Solution

The Murnaghan–Nakayama rule states that if a permutation has a

k

-cycle and remaining cycle type

ρ

, then

χ^{λ} (k, ρ) = \sum_{H} (- 1)^{leg (H)} χ^{λ ∖ H} (ρ),

where

H

runs over the removable rim hooks of length

k

. Iterating removes rim hooks whose lengths are the cycle lengths, and the character value is the signed sum over all complete removal sequences.

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