Young subgroup

In mathematics, the Young subgroups of the symmetric group ${\displaystyle S_{n}}$ are special subgroups that arise in combinatorics and representation theory. When ${\displaystyle S_{n}}$ is viewed as the group of permutations of the set ${\displaystyle \{1,2,\ldots ,n\}}$, and if ${\displaystyle \lambda =(\lambda _{1},\ldots ,\lambda _{\ell })}$ is an integer partition of ${\displaystyle n}$, then the Young subgroup ${\displaystyle S_{\lambda }}$ indexed by ${\displaystyle \lambda }$ is defined by $${\displaystyle S_{\lambda }=S_{\{1,2,\ldots ,\lambda _{1}\}}\times S_{\{\lambda _{1}+1,\lambda _{1}+2,\ldots ,\lambda _{1}+\lambda _{2}\}}\times \cdots \times S_{\{n-\lambda _{\ell }+1,n-\lambda _{\ell }+2,\ldots ,n\}},}$$ where ${\displaystyle S_{\{a,b,\ldots \}}}$ denotes the set of permutations of ${\displaystyle \{a,b,\ldots \}}$ and ${\displaystyle \times }$ denotes the direct product of groups. Abstractly, ${\displaystyle S_{\lambda }}$ is isomorphic to the product ${\displaystyle S_{\lambda _{1}}\times S_{\lambda _{2}}\times \cdots \times S_{\lambda _{\ell }}}$. Young subgroups are named for Alfred Young.^[1]

When ${\displaystyle S_{n}}$ is viewed as a reflection group, its Young subgroups are precisely its parabolic subgroups. They may equivalently be defined as the subgroups generated by a subset of the adjacent transpositions ${\displaystyle (1\ 2),(2\ 3),\ldots ,(n-1\ n)}$.^[2]

In some cases, the name Young subgroup is used more generally for the product ${\displaystyle S_{B_{1}}\times \cdots \times S_{B_{\ell }}}$, where ${\displaystyle \{B_{1},\ldots ,B_{\ell }\}}$ is any set partition of ${\displaystyle \{1,\ldots ,n\}}$ (that is, a collection of disjoint, nonempty subsets whose union is ${\displaystyle \{1,\ldots ,n\}}$).^[3] This more general family of subgroups consists of all the conjugates of those under the previous definition.^[4] These subgroups may also be characterized as the subgroups of ${\displaystyle S_{n}}$ that are generated by a set of transpositions.^[5]