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

When S n {\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 ( 1 2 ) , ( 2 3 ) , … , ( n − 1 n ) {\displaystyle (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 × ⋯ × S B ℓ {\displaystyle S_{B_{1}}\times \cdots \times S_{B_{\ell }}}, where { B 1 , … , B ℓ } {\displaystyle \{B_{1},\ldots ,B_{\ell }\}} is any set partition of { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} (that is, a collection of disjoint, nonempty subsets whose union is { 1 , … , n } {\displaystyle \{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 {\displaystyle S_{n}} that are generated by a set of transpositions.

Further reading