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I am looking for references that may formalize the following idea: let $R = k[X]$ be the coordinate ring for a generic $n \times m$ matrix $M$. It is well known that the ideal of $r \times r$ minors is invariant under the induced $GL_n \times GL_m$-action. Suppose instead that one has $2$ different groups $G$ and $H$ acting on the variables of $R$, and the ideals $I$ and $J$ that are invariant under the action by $G$ and $H$, respectively. Then $I + J$ is acted on by both $G$ and $H$, but it is not acted on by $G \times H$, mainly because the actions by $G$ and $H$ do not necessarily commute.

My question is the following: is it still possible to view $I+J$ as having an action by some other group, involving $G$ and $H$? Or, have objects with "partial" group actions (like $I+J$ as above) been studied before?

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    $\begingroup$ How about the action of $G \ast H$? $\endgroup$ Jan 23, 2021 at 19:38
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    $\begingroup$ You know what, I think that is exactly what I want. $\endgroup$
    – Rellek
    Jan 23, 2021 at 19:40
  • $\begingroup$ Is also $I$ invariant under $H$, and $J$ invariant under $G$? If not, I’m not sure if $I+J$ carries any nice action. But $IJ$ should, I think. $\endgroup$ Jan 23, 2021 at 21:16
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    $\begingroup$ There is a theory of partial group actions by Exel but it is not exactly what you are looking at $\endgroup$ Jan 23, 2021 at 21:21
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    $\begingroup$ Exel's notion is even completely opposite: all group element act, but with partial domain and codomain. $\endgroup$
    – YCor
    Jan 23, 2021 at 23:12

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