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I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in another paper of Hannabuss's with Alan Carey: Infinite dimensional groups and Riemann surface field theories.

As far as I can make out, this is the following. Suppose that $G$ is a group and $H<G$ a subgroup. (I'm being purposefully vague here about the kinds of groups: finite, Lie,...)

Let $\operatorname{Res}: \operatorname{Rep}(G) \to \operatorname{Rep}(H)$ denote the restriction functor from the category of complex $G$-modules to the category of complex $H$-modules, and let $\operatorname{Ind}: \operatorname{Rep}(H) \to \operatorname{Rep}(G)$ denote the induction functor going the other way.

Then the "permanence relation" seems to say that for every $G$-module $V$ and every $H$-module $W$,

$$\operatorname{Ind}(W) \otimes_{\mathbb{C}} V \cong \operatorname{Ind}(W \otimes_{\mathbb{C}}\operatorname{Res}(V))$$ as $G$-modules.

This strikes me as the $\otimes$-version of the following isomorphism

$$\operatorname{Hom}_H(W,\operatorname{Res}(V)) \cong \operatorname{Hom}_G(\operatorname{Ind}(W),V)$$

showing that $\operatorname{Ind}$ and $\operatorname{Res}$ are adjoint functors.

I'm not asking for a proof of the "permanence relation", which at least for finite group is not difficult, but more for a modern interpretation along the lines of the categorical interpretation of the above isomorphism of $\operatorname{Hom}$'s. And perhaps also for a more modern name by which it might be known?

I just came across another name for this formula (based Tom's comment below): apparently it's also called a "push-pull" formula and there's even an earlier MO question about it! Although that question explicitly mentions the above "permanence relation" as one of the avatars of the "push-pull" formula, none of the answers address this particular avatar. Still, if people think that this question is a duplicate, I will not be offended if it is closed as such!

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# What is the "permanence relation" really?

I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in another paper of Hannabuss's with Alan Carey: Infinite dimensional groups and Riemann surface field theories.

As far as I can make out, this is the following. Suppose that $G$ is a group and $H<G$ a subgroup. (I'm being purposefully vague here about the kinds of groups: finite, Lie,...)

Let $\operatorname{Res}: \operatorname{Rep}(G) \to \operatorname{Rep}(H)$ denote the restriction functor from the category of complex $G$-modules to the category of complex $H$-modules, and let $\operatorname{Ind}: \operatorname{Rep}(H) \to \operatorname{Rep}(G)$ denote the induction functor going the other way.

Then the "permanence relation" seems to say that for every $G$-module $V$ and every $H$-module $W$,

$$\operatorname{Ind}(W) \otimes_{\mathbb{C}} V \cong \operatorname{Ind}(W \otimes_{\mathbb{C}}\operatorname{Res}(V))$$ as $G$-modules.

This strikes me as the $\otimes$-version of the following isomorphism

$$\operatorname{Hom}_H(W,\operatorname{Res}(V)) \cong \operatorname{Hom}_G(\operatorname{Ind}(W),V)$$

showing that $\operatorname{Ind}$ and $\operatorname{Res}$ are adjoint functors.

I'm not asking for a proof of the "permanence relation", which at least for finite group is not difficult, but more for a modern interpretation along the lines of the categorical interpretation of the above isomorphism of $\operatorname{Hom}$'s. And perhaps also for a more modern name by which it might be known?