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To be specific: if $G$ is a finite group and $\operatorname{Irr}(G)$ the set of its irreducible characters (over the complex field) then we know that $$ 1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^2}{|G|} $$ where $d_\phi=\phi(e)$ is the degree of the character $\phi$. Thus we get a probability measure on $\operatorname{Irr}(G)$ which assigns mass $|G|^{-1}(d_\phi)^2$ to the atom $\{\phi\}$.

My question is: among finite group theorists, what is the usual name for this measure? I need to know since I am writing a paper which might include this audience, and I don't wish to clash with the accepted conventions unless it's necessary.

I think that in the context of e.g. random Young diagrams and representations of the symmetric group, this is sometimes referred to as "Plancherel measure". However, for a non-commutative harmonic analyst, it is traditional and more natural to use "Plancherel measure" for the measure $\nu$ on the dual of a (Type I, unimodular) locally compact group $G$ which makes the Plancherel formula hold $$ \int_G \lvert f(x)\rvert^2 \,dx = \int_{\widehat{G}} \lVert \pi(f) \rVert_{\rm HS}^2\,d\nu(\pi) $$ Note that with this convention, if $G$ is finite then $\nu(\{\phi\})=|G|^{-1} d_\phi$, and so $\nu$ is usually not a probability measure.

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    $\begingroup$ I've always seem Plancherel measure. Wiki agrees en.m.wikipedia.org/wiki/Plancherel_measure $\endgroup$ May 2, 2022 at 15:13
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    $\begingroup$ It never occurred to me before reading that Wikipedia page that the Plancherel measure was named after Plancherel directly. I always assumed there was some deeper connection to some type of non-commutative Plancherel's/Parseval's theorem (is there?). $\endgroup$ May 2, 2022 at 15:28
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    $\begingroup$ Thanks @BenjaminSteinberg - well, I guess that is what I will have to use. This particular paper probably only deals with finite groups. $\endgroup$
    – Yemon Choi
    May 2, 2022 at 15:36
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    $\begingroup$ Although as @LSpice has pointed out, the wikipedia page is not consistent since the Plancherel measure they mention for compact groups is the "harmonic analyst's Plancherel measure" and differs from the definition they give above for finite groups. (Note that the difference cannot be rescued by rescaling Haar measure) $\endgroup$
    – Yemon Choi
    May 2, 2022 at 15:38
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    $\begingroup$ @YemonChoi I think it’s not so much that Wikipedia is inconsistent, but that the literature is inconsistent (in a way that is not uncommon to see in many emerging areas of research). But my impression — admittedly biased since I work in combinatorial probability — is that almost all the work on Plancherel measures has been in the context of finite groups, and almost all of that was on the specific case of the symmetric group. So as long as you make clear the context and include precise definitions, I think “Plancherel measure” is indeed the best term to use. $\endgroup$
    – Dan Romik
    May 2, 2022 at 18:26

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