Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this bijection natural?
This question is somewhat informal, however I hope it can be left in such a form. One of the ways to make it formal is to ask about the properties which distinguish the bijection above among all bijections between the two sets. E.g. properties like: identity class is mapped to trivial irreps...
It is very standard material that for any finite group the number of conjugacy classes and irreps is the same, however it is striking that no natural bijection between the two sets is known (and may not exist at all).
MathOverflow has already several discussions around the subject, however they are different from the present question. A few of them are listed below.
Bijection between irreducible representations and conjugacy classes of finite groups discusses bijections for general groups.
The symmetric groups are not the only groups with a "natural" bijection. Some examples are collected here: Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?
Related questions: G/[G,G], irreps and conjugacy classes, Duality between conjugacy classes and irreducible characters for finite monoids?, among others.
Conjugacy classes and irreps have many similar properties. However, in general, they do not fit each other exactly: Action of Out(G) on both sets, "reality properties" (Are there “real” vs. “quaternionic” conjugacy classes in finite groups?, If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?).