nLab Young subgroup


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


Let n1n \ge 1. Given a composition (λ 1,...,λ k)(\lambda_{1},...,\lambda_{k}) of the integer nn (where λ i1\lambda_{i} \ge 1), we have the Young subgroup:

𝔖 λ 1×...×𝔖 λ k𝔖 n\mathfrak{S}_{\lambda_{1}} \times ... \times \mathfrak{S}_{\lambda_{k}} \hookrightarrow \mathfrak{S}_{n}

associated to this composition.


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

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

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

symmetric group

Last revised on July 30, 2022 at 20:22:41. See the history of this page for a list of all contributions to it.