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.

symmetric group

Last revised on July 30, 2022 at 20:22:41.
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