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nLab > Latest Changes: wreath product of groups

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  1.  
    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJul 18th 2024

    Finally, some classical references added. Category class algebra added.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025
    • (edited Jan 25th 2025)

    made a bunch of cosmetic edits to this entry (still leaving room for more)

    The section on “relation to the Borel construction” still needs an argument or a citation

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025
    • (edited Jan 25th 2025)

    In fact, the previous statement about Borel constructions (from revision 3 in 2015) seems to be not even wrong to me, even with the notation fixed. (For the trivial G-action it’s true, if one also specifies that the G-permutation action is the trivial one on the singleton, but that case is besides the point of wreath products.)

    So I have deleted it. But let me know if I am missing something.

    Instead I have added (as this example) the statement that

    π1(Xn⫽Sym(n))≃π1(X)≀Sym(n),

    which is possibly what the original author actually had in mind.

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025

    Also added a more straightforward formulation of the definition.

    Kept the previous version, equipped now with a pointer to Holland 1989, who puts it that way.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025

    added pointer to:

    • Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann: Wreath products, Chapter 8 of: Notes on Infinite Permutation Groups, Lecture Notes in Mathematics 1698, Springer (2006) 67-76 [doi:10.1007/BFb0092558]

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025

    added a remark on notation (here)

    and moved the previous definition to a remark “As a permutation group” under “Properties” (now here)

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2025
    • (edited Jan 25th 2025)

    and finally I added what I actually came for to this entry, namely references on the representation theory of wreath products with symmetric groups:

    • Gordon D. James, Adalbert Kerber: Representations of Wreath Products, Chapter 4 of: The Representation Theory of the Symmetric Group, Cambridge University Press (1984) [doi:10.1017/CBO9781107340732]

    • I. A. Pushkarev: On the representation theory of wreath products of finite groups and symmetric groups, Journal of Mathematical Sciences, 96 (1999) 3590–3599 [doi:10.1007/BF02175835]

      originally in: Representation theory, dynamical systems, combinatorial and algorithmic methods. Part II, Zap. Nauchn. Sem. POMI 240, POMI, St. Petersburg (1997) 229–244 [mathnet:znsl475]

    Namely, I want to understand the representation theory of ℤk≀Sym(n). But not tonight, have an early flight tomorrow morning…

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2025

    added mentioning of the terminology “complete monomial groups” for the wreath products of the form H≀Sym(n) with Sym(n) understood with its defining action.

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2025
    • (edited Jan 26th 2025)

    Just found this, which is a similar story to what I am after:

    On wreath products of a cyclic with a symmetric group as analogs of (anyon) braid groups in dimension >2:

    diff, v10, current

1 to 9 of 9

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