---
tag: E86KAZ914F
kind: concept
name: Young subgroup
sources:
  - wikipedia: en:76019869@r1344816453
    locator: Young subgroup
    note: adopted
    mention: wikipedia://en/76019869
    sections: [Summary]
same_as:
  wikidata: Q124625185
  wikipedia: Young subgroup
inventory: [64R5WNGR1H]
---

## Summary

In mathematics, the **Young subgroups** of the [[symmetric group|0515A6D2Y1]] $S_n$ are special subgroups that arise in [[combinatorics|0JKW7W7XFR]] and representation theory. When $S_n$ is viewed as the group of [[permutations|05RWFKVTJ7]] of the set $\{1, 2, \ldots, n\}$, and if $\lambda = (\lambda_1, \ldots, \lambda_\ell)$ is an [[integer partition|0JV30XN73Z]] of $n$, then the Young subgroup $S_\lambda$ indexed by $\lambda$ is defined by

$$S_\lambda = S_{\{1, 2, \ldots, \lambda_1\}} \times S_{\{\lambda_1 + 1, \lambda_1 + 2, \ldots, \lambda_1 + \lambda_2\}} \times \cdots \times S_{\{n - \lambda_\ell + 1, n - \lambda_\ell + 2, \ldots, n\}},$$

where $S_{\{a, b, \ldots\}}$ denotes the set of permutations of $\{a, b, \ldots\}$ and $\times$ denotes the [[direct product of groups|00HF4RJ5QN]]. Abstractly, $S_\lambda$ is isomorphic to the product $S_{\lambda_1} \times S_{\lambda_2} \times \cdots \times S_{\lambda_\ell}$. Young subgroups are named for Alfred Young.