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I am interested in any literature about the following mathematical property.

Let $V$ a vector space and $G$ a group acting on $V$.

What is the name for the property of a set of operators $H=\{h:V\to V\}$ being stable under conjugation by $G$:

$$ gHg^{-1}=H,\;\;\; \forall g\in G $$

i.e. the action of each element of $G$ on $H$ is a permutation of the $H$ elements?

Does this property have a name? Has this property been studied, even for particular groups?

Thank you.

Fabio

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    $\begingroup$ do I understand this correctly? take as an example for $G$ the group of orthogonal matrices, then the operators $H$ are multiplication by a scalar. $\endgroup$
    – Carlo Beenakker
    Commented Apr 29, 2022 at 20:50
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    $\begingroup$ That condition is circular: you can't test whether $g h g^{-1}$ belongs to $H$ until you've defined $H$, so you can't use that requirement in the definition of $H$. $\endgroup$
    – LSpice
    Commented Apr 29, 2022 at 21:59
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    $\begingroup$ I think Fabio is asking for a name for the property of a subset $H$ of linear endomorphism of $V$ being closed under conjugation by $G$. $\endgroup$
    – Sam Hopkins
    Commented Apr 29, 2022 at 22:02
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    $\begingroup$ Notice that if $H$ has that property, so does its linear span. After this extension, you are basically looking at $G$-invariant subspaces of ${\rm End}(V)$. $\endgroup$
    – Geoff Robinson
    Commented Apr 30, 2022 at 7:32
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    $\begingroup$ Could anybody who understood the OP's intent edit the question so as to make correctly stated? $\endgroup$
    – YCor
    Commented Apr 30, 2022 at 8:24

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