In mathematics, the Young subgroups of the symmetric group are special subgroups that arise in combinatorics and representation theory. When is viewed as the group of permutations of the set , and if is an integer partition of , then the Young subgroup indexed by is defined by
where denotes the set of permutations of and denotes the direct product of groups. Abstractly, is isomorphic to the product . Young subgroups are named for Alfred Young.[1]
When 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 .[2]
In some cases, the name Young subgroup is used more generally for the product , where is any set partition of (that is, a collection of disjoint, nonempty subsets whose union is ).[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 that are generated by a set of transpositions.[5]