Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S\_n$)?



Bijection between irreducible representations and conjugacy classes of finite groups
This is a different take on Steven Landsburg's answer. The short version is that conjugacy classes and irreducible representations should be thought of as being dual to each other. 


Fix an algebraically closed field $k$ of characteristic not dividing the order of our finite group $G$. The group algebra $k[G]$ is a finite-dimensional Hopf algebra, so its dual is also a finite-dimensional Hopf algebra of the same dimension; it is the Hopf algebra of functions $G \to k$, which I will denote by $C(G)$. (The former is cocommutative but not commutative in general, while the latter is commutative but not cocommutative in general.) The dual pairing
$$k[G] \times C(G) \to k$$


is equivariant with respect to the action of $G$ by conjugation, and it restricts to a dual pairing
$$Z(k[G]) \times C\_{\text{cl}}(G) \to k$$


on the subalgebras fixed by conjugation; $Z(k[G])$ is the center of $k[G]$ and $C\_{\text{cl}}(G)$ is the space of class functions $G \to k$. Now:



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> The maximal spectrum of $Z(k[G])$ can be canonically identified with the irreducible representations of $G$, and the maximal spectrum of $C\_{\text{cl}}(G)$ can be canonically identified with the conjugacy classes of $G$.
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The second identification should be clear; the first comes from considering the central character of an irreducible representation. Now, the pairing above is nondegenerate, so to every point of the maximal spectrum of $Z(k[G])$ we can canonically associate an element of $C\_{\text{cl}}(G)$ (the corresponding irreducible character) and to every point of the maximal spectrum of $C\_{\text{cl}}(G)$ we can canonically associate an element of $Z(k[G])$ (the corresponding sum over a conjugacy class divided by its size). 


