Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?
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For most finite groups, the bijection between conjugacy classes and irreducible $\mathbb{C}$-representations is not in any sense (known to me, at least) canonical. One sees this already for finite abelian groups: the isomorphism between such a group $G$ and its character group is non-canonical, and this at least can be formalized and proved using category theory.
But yes, there are examples. For finite symmetric groups, this is achieved via the theory of Young diagrams. See this wikipedia article for a brief introduction and, if you like, consult the references given for more information.
answered Jun 8, 2011 at 23:48
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$\begingroup$ Why can the embedding of a finite group into $S_n$ not be used here? $\endgroup$ Commented Jun 9, 2011 at 6:35
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2$\begingroup$ @user11848: I'm afraid I don't understand your question. Knowing the irreducible representations of a group does not tell you the irreducible representations of a subgroup in any obvious way. $\endgroup$ Commented Jun 9, 2011 at 6:50
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$\begingroup$ yes, in general that might be a hard task, but what if it is a normal subgroup? Does the Mackey machine not give a construction here? $\endgroup$ Commented Jun 9, 2011 at 12:08
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$\begingroup$ @PeteL.Clark What is the list of other examples known ? Where can I take it ? I am googling for quite a long, but do not see comprehensive answer :( $\endgroup$ Commented Aug 13, 2012 at 10:32