Young subgroup

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Let $\{1,2,\ldots,n\} = \cup_{i=1}^k \alpha_i$ be a partition of $\{1,2,\ldots,n\}$ into $k$ disjoint subsets. Then the corresponding Young subgroup of $S_n$, the symmetric group on $n$ letters, is the subgroup $$ S_{\alpha_1} \times \cdots \times S_{\alpha_k} \,, $$ where $S_{\alpha_i} = \{ \sigma \in S_n : \sigma(j) = j \ \text{for all}\ j \not\in \alpha_i \}$. Sometimes only the particular cases $$ S_{\alpha_1} \times \cdots \times S_{\alpha_k} $$ are meant where $\alpha_i = \{\lambda_{i-1} + 1,\ldots, \lambda_i\}$, where $\lambda_0 = 0$ and $\lambda = (\lambda_1,\ldots,\lambda_k)$ is a partition of the natural number $n$, i.e. $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$, $\sum \lambda_i = n$.


[a1] G.D. James, "The representation theory of the symmetric groups" , Springer (1978) pp. 13
[a2] A. Kerber, "Representations of permutation groups" , I , Springer (1971) pp. 17
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