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My question is the following: is there a (call it "canonical", "standard", or some other interesting and known) representation (probably reducible) of a finite group, which dimension is equal to a number of conjugacy classes of the group? It can be complex, and can even be projective. My main interest is the case of a symmetric group.

I am not interested in trivial cases like a direct sum of many trivial representations, for example.

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    $\begingroup$ I think this is very unlikely. For instance $\mathrm{PSL}_2(\mathbb{F}_p)$ has $(p+5)/2$ conjugacy classes and its irreducible complex characters have dimensions $1$, $(p\pm 1)/2$ (twice, sign depending on $p$ modulo $4$), $p-1$, $p$, $p+1$. So at least $2$ copies of the trivial character are needed. $\endgroup$
    – Mark Wildon
    Commented Nov 19, 2022 at 15:16
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    $\begingroup$ A more natural question is a representation whose dimension is $k(G)\cdot |G|$, where $k(G)$ is the number of conjugacy classes. Such a representation has character $\psi(w)=\#\{(u,v)\in G\times G\,:\, w=uvu^{-1}v^{-1}\}$. Moreover, the multiplicity of an irreducible character $\chi$ is $|G|/\dim(\chi)$ (a positive integer). It is an open problem to find a "natural" action of $G$ that affords this representation. This would provide a new proof that $\dim(\chi)$ divides $|G|$. $\endgroup$
    – Richard Stanley
    Commented Nov 19, 2022 at 15:47
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    $\begingroup$ And for the monster group, the smallest degree of a nontrivial representation is way bigger than the number of conjugacy classes. $\endgroup$
    – Jeremy Rickard
    Commented Nov 19, 2022 at 16:10
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    $\begingroup$ What would be the purpose or interest of such a representation? \\ @RichardStanley, I am confused by "Such a representation has character …". Surely this character formula does not follow just from the dimension? Do you mean rather that one looks for a representation with that character, whose dimension is then necessarily $k(G)\cdot\lvert G\rvert$? $\endgroup$
    – LSpice
    Commented Nov 19, 2022 at 17:41
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    $\begingroup$ @LSpice: I should have said "An example of such a representation ..." $\endgroup$
    – Richard Stanley
    Commented Nov 20, 2022 at 3:25

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