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.^[1]

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 $(1\ 2),(2\ 3),\ldots ,(n-1\ n)$ .^[2]

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 $\{1,\ldots ,n\}$ ).^[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 $S_{n}$ that are generated by a set of transpositions.^[5]

References

^ Sagan, Bruce (2001), The Symmetric Group (2 ed.), Springer-Verlag, p. 54
^ Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, Springer, p. 41, doi:10.1007/3-540-27596-7, ISBN 978-3540-442387
^ Kerber, A. (1971), Representations of permutation groups, vol. I, Springer-Verlag, p. 17
^ Jones, Andrew R. (1996), "A Combinatorial Approach to the Double Cosets of the Symmetric Group with respect to Young Subgroups", European Journal of Combinatorics, 17 (7): 647–655, doi:10.1006/eujc.1996.0056
^ Douvropoulos, Theo; Lewis, Joel Brewster; Morales, Alejandro H. (2022), "Hurwitz Numbers for Reflection Groups I: Generatingfunctionology", Enumerative Combinatorics and Applications, 2 (3): Article #S2R20, arXiv:2112.03427, doi:10.54550/ECA2022V2S3R20

References

Further reading