Past exam of the representation Theory of Symmetric Groups course of the University of Oxford / 2011 / 1 / c

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- Solution c

Solution

The Young subgroup for

(1^{n})

is trivial. Hence

M_{C}^{(1^{n})}

is the regular representation of

S_{n}

. The subgroup-average formula reduces to

m_{(1^{n}), μ} = χ_{μ} (1) = dim S_{C}^{μ} = \frac{n !}{\prod _{u \in [μ]} h ( u )} .

This is the usual multiplicity of an irreducible representation in the regular representation.

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