# Young subgroup

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. $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$, $\sum \lambda_i = n$.

#### References

[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 |

**How to Cite This Entry:**

Young subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Young_subgroup&oldid=41926