# 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?

Arunas Liulevicius, in his "Arrows, Symmetries and Representation Rings", interprets $H$-modules as $G$-vector bundles over the discrete $G$-manifold (i. e., $G$-set) $G\diagup H$. Restriction is pullback and induction is pushforward (defined for a finite covering). This way, I guess, the permanence relation becomes the projection formula for vector bundles - which, however, is not less mysterious to me... –  darij grinberg Jul 29 '10 at 19:29
More precisely, a $G$-vector bundle over a discrete $G$-manifold $M$ (this simply means a $G$-set with $G$ acting from the left) means a family $\left(V_m\right)_{m\in M}$ of vector spaces $V_m$ of $G$ for each $m\in M$ equipped with maps $t_{m,g}:V_m\to V_{gm}$ for each $m\in M$ and $g\in G$ satisfying $t_{gm,h}t_{m,g}=t_{m,hg}$ for any $m\in M$, $g\in G$ and $h\in H$. Of course, when $M$ is the trivial $1$-point $G$-set, then there is only one $V_m$ and thus a $G$-vector bundle is just a representation of $G$. Generally, when $M$ is the $G$-set $G\diagup H$, ... –  darij grinberg Jul 29 '10 at 19:37
... then the $G$-vector bundle is uniquely determined (up to isomorphism) by the vector space $V_1$ and the maps $t_{h,1}$ for $h\in H$, so it is more or less the same as a representation of $H$. –  darij grinberg Jul 29 '10 at 19:38
The equation expressing what might be called the "naturality of the cap product" (pulling back the cohomology class $a$, capping it with the homology class $b$, pushing it forward gives the same result as pushing $b$ forward and capping with $a$) seems to have the old-fashioned name "the permanence formula". At least, that is what Bott called it when I was learning algebraic topology from him in the 1970s. –  Tom Goodwillie Jul 29 '10 at 21:06