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 $$\{1, 2, \ldots, n\}$$, and if $$\lambda = (\lambda_1, \ldots, \lambda_\ell)$$ is an integer partition of $$n$$, then the Young subgroup $$S_\lambda$$ indexed by $$\lambda$$ is defined by $$S_\lambda = S_{\{1, 2, \ldots, \lambda_1\}} \times S_{\{\lambda_1 + 1, \lambda_1 + 2, \ldots, \lambda_1 + \lambda_2\}} \times \cdots \times S_{\{n - \lambda_\ell + 1, n - \lambda_\ell + 2, \ldots, n\}},$$ where $$S_{\{a, b, \ldots\}}$$ denotes the set of permutations of $$\{a, b, \ldots\}$$ 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 $$(1 \ 2), (2 \ 3), \ldots, (n - 1 \ n)$$.

In some cases, the name Young subgroup is used more generally for the product $$S_{B_1} \times \cdots \times S_{B_\ell}$$, 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.