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I've recently realized that there is an isomorphism between the integer partitions of $n$ and the symmetric group $S_n$. There are many documented properties of integer partitions of $n$ but I can't seem to find much similar prior work on conjugacy classes of $S_n$. I'm hoping to use these properties of the conjugacy classes to find some insights on some research I'm doing.

I've spent a couple of days trying to define invariants myself but can't seem to produce many that are useful.

I have a feeling this is work that's been explored before but I don't have the correct vocabulary to describe it well enough to find it. Or maybe I'm taking the wrong approach entirely. Can someone please help me out? Thanks!

(The partition research in question is here)

EDIT: I'm adding this to hopefully make this question more clear:

Regardless of everything I've mentioned, all I'm looking for is properties of conjugacy classes formed in $S_n$ by cycle type.

For instance, partitions of integer $n$ have many studied properties such as durfee square, rank, crank, etc...

But I can't seem to find any information on any useful/nontrivial properties of conjugacy classes in the corresponding group $S_n$. I'm hoping that someone more experienced can help point me to useful properties as I can't seem to find much prior art. Thanks!

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  • $\begingroup$ Exactly what is the isomorphism you've recently realized? $\endgroup$
    – Berci
    Commented Dec 24, 2019 at 19:59
  • $\begingroup$ Welcome to Mathematics Stack Exchange. Cycle type determines conjugacy class $\endgroup$
    – J. W. Tanner
    Commented Dec 24, 2019 at 20:06
  • $\begingroup$ @Berci, I've added the link to an example of the isomorphism I'm describing. I haven't written or read a formal proof of the isomorphism but both sets are of the same cardinality and there's a 1:1 mapping between the "parts" of size $q$ in a given partition and the number of $q-cycles$ in the symmetric group. You can see that here: groupprops.subwiki.org/wiki/… $\endgroup$
    – Brian
    Commented Dec 24, 2019 at 20:12
  • $\begingroup$ @J.W.Tanner, thanks for the link! I think I might not have been clear enough. I'm aware that the cycle type of a permutation in the symmetric group defines what partition it should be sent to (i.e. $(1, 3, 5)(2, 4)(6)(7, 8)$ has cycle type $(3,2,1,2)$ which forms the mapping to the partition: $(3,2,2,1)$). But I'm looking for properties of the conjugacy group that $(1, 3, 5)(2, 4)(6)(7, 8)$ belongs to in $S_8$ (which we can call $(1, 2, 3)(4, 5)(6, 7)$ for simplicity. $\endgroup$
    – Brian
    Commented Dec 24, 2019 at 20:27
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    $\begingroup$ That should not be called an "isomorphism", because there is no structure that you are preserving. Rather, there is a bijective between the conjugacy classes of $S_n$ and the partitions of $n$, in that each partition determines a conjugacy class and vice-versa, and these relations are inverses of each other. But when you talk about "isomorphism" in the context of group theory, you are implicitly talking about a bijection that preserves the group structure. That's not what we have here. $\endgroup$
    – Arturo Magidin
    Commented Dec 24, 2019 at 20:27

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