nLab Young subgroup

Idea

Every composition of an integer $n$ induces a subgroup of $\mathfrak{S}_{n}$.

Definition

Let $n \ge 1$. Given a composition $(\lambda_{1},...,\lambda_{k})$ of the integer $n$ (where $\lambda_{i} \ge 1$), we have the Young subgroup:

$\mathfrak{S}_{\lambda_{1}} \times ... \times \mathfrak{S}_{\lambda_{k}} \hookrightarrow \mathfrak{S}_{n}$

associated to this composition.

Example

If $(k,1,l)$ is a composition of $n$ as above, then:

$\mathfrak{S}_{k} \times \mathfrak{S}_{1} \times \mathfrak{S}_{l} \hookrightarrow \mathfrak{S}_{n}$

is a subgroup of permutations $\sigma \in \mathfrak{S}_{n}$ which verify that $k+1$ is a fixed point of $\sigma$. However some permutations fix $k+1$ without being an element of this subgroup.

Related notions

symmetric group