In mathematics, the Young subgroups of the symmetric group
are special subgroups that arise in combinatorics and representation theory.
is viewed as the group of permutations of the set
ℓ
is an integer partition of
, then the Young subgroup
indexed by
is defined by
ℓ
ℓ
denotes the set of permutations of
denotes the direct product of groups.
Abstractly,
is isomorphic to the product
ℓ
Young subgroups are named for Alfred Young.
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
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.