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I have often heard that Hilbert killed invariant theory, but I see that there computational invariant theory seems to be an active field, and I understand that geometric invariant theory arose from algebraic invariant theory.

What else is going on with algebraic invariant theory?

Are there contemporary surveys on open problems and the current focus of algebraic invariant theory? Is most research in the computational aspects?

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    $\begingroup$ There is a recent book by De Concini and Procesi with some fairly recent results in it. Usually the question is to find generators and relations for the invariant rings of a family of groups acting on a family of spaces (e.g., $\operatorname{GL}_n$ acting on $K^{n\times n}$ by conjugation). This is not a computational problem, since the families are typically infinite (all $n$, not just a few). Also, fields of positive characteristic offer a challenge in many situation where the characteristic-$0$ theory can be derived from general facts. $\endgroup$
    – darij grinberg
    Commented Sep 15 at 18:03
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    $\begingroup$ Also I expect a lot of activity to follow in the anticommutative and noncommutative settings (which lag behind the commutative one for historical reasons). $\endgroup$
    – darij grinberg
    Commented Sep 15 at 18:04
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    $\begingroup$ For an example of open problem, see my article mentioned at the end of my MO answer mathoverflow.net/questions/286872/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 15 at 19:59
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    $\begingroup$ A fairly recent survey article appeared in arxiv.org/abs/2403.12709 $\endgroup$
    – Gregor Kemper
    Commented Sep 16 at 6:14
  • $\begingroup$ The work of Andrew Snowden and his collaborators is worth looking at. See for example "The ideal of relations for the ring of invariants of $n$ points on the line" and other work in a similar vein. $\endgroup$
    – Timothy Chow
    Commented Sep 16 at 11:58

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