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I always found it strange that, in the context of invariant and representation theory, averaging over the group is called the "Reynolds operator". As far as I know the work of Reynolds was in fluid mechanics. He introduced the idea of local averaging in order to distinguish slow and fast variables. As such it is a precursor of things such as the Lyapunov-Schmidt method and Wilson's renormalization group. How did this terminology end up in invariant theory?

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The history is probably hard to reconstruct in modern terms, given the early dates of papers in invariant theory. Anyway it does seem certain that Reynolds himself was working in other directions, as the incomplete Wikipedia article suggests: – Jim Humphreys Aug 11 '13 at 13:29
P.S. If you can track down the conference article by Rota listed in the references on Wikipedia, that might be helpful. Rota was deeply interested in both invariant theory and history, though he could be polemical at times. – Jim Humphreys Aug 11 '13 at 13:34
@Jim: Thanks. I don't have access now to a math library but from the few pages of Rota's article I could see on Google Books it seems that he might be responsible for this terminology. As per your earlier comment, indeed it is not so easy to reconstruct that history. It is a bit like doing archeological excavations. Also historical development does not follow the logical one. Namely, the logical order of things would seem to me to be: 1- realizing the usefulness of group averaging in invariant theory (eg as in Hibert's proof of finite generation), ... – Abdelmalek Abdesselam Aug 11 '13 at 13:54
...2- constructing such an averaging or Haar integral for compact groups such as SO(n) and SU(n), 3- implementing this integration in a more algebraic way, e.g. as a differential operator involving Cayley's Omega operator, which is the analogue of expressing a Gaussian integral via the heat kernel. The funny thing is: 3 was used by Clebsch in 1861 to prove the FFT for SL(n), 2 was done by Hurwitz in 1897, while 1 i.e. Hilbert's proof which uses 3 goes back to 1890. – Abdelmalek Abdesselam Aug 11 '13 at 14:04
Dumb question, but is it obvious that both are due to the same Reynolds? It's a common name... – Nate Eldredge Aug 11 '13 at 23:21

I recall this old article in German by Hilbert 1890: Hilbert, David (1890), "Über die Theorie der algebraischen Formen." ( where in proof of his theorems on invariant theory he applies the Reynolds operator implying Cayley's omega process. The article is unfortunately in German and I encountered that years ago, irritated as a process engineer who was into fluid mechanics, why Reynolds does into such math stuff. So far I can only answer your question from the other side of the border as engineer, how Hilbert got to Reynolds. I just found also a weak snapshot on this on wiki not that extensive: Nevertheless I think some further reconstruction work would be surely required on illuminating all historic details.

Append moved from my comments: one question however as your title is bit irritating, you are not looking for the authentic name/term Reynolds Operator but the mathematics behind of it, isnt it? Because then one should see back into the history of the term Reynolds Operator and that might be not perfectly synchronous to your time table above or my answer. The name/term Reynolds Operator itself must have been plugged in perhaps later (possibly parallel to Heavyside 1880-87 systematic operational calculus or even later).

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WTF a 1890 paper not in open access? Anyway, here is a working link: – darij grinberg Aug 11 '13 at 16:55

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