Young subgroup

In mathematics, the Young subgroups of the symmetric group $S_{n}$ are special subgroups that arise in combinatorics and representation theory. When $S_{n}$ is viewed as the group of permutations of the set $\{1,2,\ldots ,n\}$, and if $\lambda =(\lambda _{1},\ldots ,\lambda _{\ell })$ is an integer partition 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. 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.

When $S_{n}$ 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 $(1\ 2),(2\ 3),\ldots ,(n-1\ n)$.

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

Further reading

• Borevich, Z.I.; Gavron, P.V. (1985), "Arrangement of Young subgroups in the symmetric group", Journal of Soviet Mathematics, 30: 1816–1823, doi:10.1007/BF02105094
• "Young subgroup", Encyclopedia of Mathematics, EMS Press, 2001 [1994]