# Is there a theory of "partial" group actions?

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?

• How about the action of $G \ast H$? Jan 23, 2021 at 19:38
• You know what, I think that is exactly what I want. Jan 23, 2021 at 19:40
• 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. Jan 23, 2021 at 21:16
• There is a theory of partial group actions by Exel but it is not exactly what you are looking at Jan 23, 2021 at 21:21
• Exel's notion is even completely opposite: all group element act, but with partial domain and codomain.
– YCor
Jan 23, 2021 at 23:12