Talk:Chevalley–Shephard–Todd theorem

Serre

"It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre." Well, okay, that's literally true, but it does a pretty severe disservice to the actual context, and is a little vague about whether "it" refers to Chevalley's result or Shephard--Todd's as well.

Chevalley's paper was in fact written by Armand Borel without Chevalley's direct knowledge---see Mark Goresky's web page for a retelling. Chevalley had in mind the characteristic 0 case, but Borel wanted to use the result in the non-modular case. Chevalley wasn't interested in bothering to publish the result, but Borel got his notes and made a short paper out of the argument, slightly tweaking the exposition to cover the non-modular case and with a reference to Borel's own forthcoming paper as the only citation. Chevalley's result as written up by Borel showed that finite groups generated by reflections (i.e. they're in particular order two) over a non-modular field have polynomial invariants. Shephard--Todd showed the converse holds for finite groups generated by pseudoreflections (i.e. they may have order larger than two) over the complex numbers. Serre noted that Chevalley's arguments work essentially word-for-word for finite groups generated by pseudoreflections in the non-modular case. Actually, Shephard--Todd's arguments work in the non-modular case as well except for an appeal to the Jacobian criterion. I know this is easy to overcome, at least---Stanley does so with an appeal to Hall's marriage theorem in his Invariants of Finite Groups paper. In any case, in my experience when people add Serre's name to this theorem, they say in the next breath that he noted Chevalley's argument works for pseudoreflections. Sadly I don't know of any of the books on the topic which have included this history formally in a way that could be cited. 2607:F720:1901:1001:137:110:37:129 (talk) 00:25, 23 July 2019 (UTC)