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 $\{1,2,\dots ,n\}$, and if $\lambda =({\lambda }_1,\dots ,{\lambda }_{\ell })$ is an integer partition of $n$, then the Young subgroup $S_{\lambda }$ indexed by $\lambda$ is defined by $$S_{\lambda }=S_{\{1,2,\dots ,{\lambda }_1\}}\times S_{\{{\lambda }_1+1,{\lambda }_1+2,\dots ,{\lambda }_1+{\lambda }_2\}}\times \cdots \times S_{\{n-{\lambda }_{\ell }+1,n-{\lambda }_{\ell }+2,\dots ,n\}},$$ where $S_{\{a,b,\dots \}}$ denotes the set of permutations of $\{a,b,\dots \}$ 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 $(12),(23),\dots ,(n-1n)$.

In some cases, the name Young subgroup is used more generally for the product $S_{B_1}\times \cdots \times S_{B_{\ell }}$, where $\{B_1,\dots ,B_{\ell }\}$ is any set partition of $\{1,\dots ,n\}$ (that is, a collection of disjoint, nonempty subsets whose union is $\{1,\dots ,n\}$). 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 
• Hazewinkel, Michiel, ed. (2001), "Young subgroup", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page