 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2007/b/7/c/solution

For type

(1^{n})

, all labels are distinct, so a semistandard tableau is precisely a standard Young tableau. Thus

K_{λ, (1^{n})} = f^{λ}

M^{(1^{n})} ≅ λ ⊢ n ⨁ f^{λ} S^{λ}, m_{λ} = f^{λ} = \frac{n !}{\prod _{c \in λ} h ( c )} .

This is also the decomposition of the regular representation, since tabloids with one entry per row are naturally indexed by all permutations.

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