Young subgroup explained

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 λ = ( λ _{1,} \ldots, λ _{\ell)} is an integer partition of n , then the Young subgroup S _{λ} indexed by λ is defined by$$S\_\backslash lambda\; =\; S\_\; \backslash times\; S\_\; \backslash times\; \backslash cdots\; \backslash times\; S\_,$$where S _{\{a,} } denotes the set of permutations of \{a, b, \ldots\} and x denotes the direct product of groups. Abstractly, S _{λ} is isomorphic to the product S _{λ _{1}} x S _{λ _{2}} x … x S _{λ _{\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}} x … x 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