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Young subgroup

From Wikipedia, the free encyclopedia

In mathematics, the Young subgroups of the symmetric group are special subgroups that arise in combinatorics and representation theory. When is viewed as the group of permutations of the set , and if is an integer partition of , then the Young subgroup indexed by is defined by where denotes the set of permutations of and denotes the direct product of groups. Abstractly, is isomorphic to the product . Young subgroups are named for Alfred Young.[1]

When 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 , where 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.[5]

References

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  1. ^ Sagan, Bruce (2001), The Symmetric Group (2 ed.), Springer-Verlag, p. 54
  2. ^ Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, Springer, p. 41, doi:10.1007/3-540-27596-7, ISBN 978-3540-442387
  3. ^ Kerber, A. (1971), Representations of permutation groups, vol. I, Springer-Verlag, p. 17
  4. ^ Jones, Andrew R. (1996), "A Combinatorial Approach to the Double Cosets of the Symmetric Group with respect to Young Subgroups", Europ. J. Combinatorics, 17: 647–655
  5. ^ 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

Further reading

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