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. |
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