What is the "permanence relation" really? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:13:36Z http://mathoverflow.net/feeds/question/33828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33828/what-is-the-permanence-relation-really What is the "permanence relation" really? José Figueroa-O'Farrill 2010-07-29T19:04:19Z 2010-07-30T00:04:50Z <p>I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss <a href="http://iopscience.iop.org/0022-3689/2/3/005" rel="nofollow"><em>The Dirac equation in de Sitter space</em></a>. The only other similar google hit for this phrase appears in another paper of Hannabuss's with Alan Carey: <a href="http://www.springerlink.com/content/y25kg0220p86m775/" rel="nofollow"><em>Infinite dimensional groups and Riemann surface field theories</em></a>.</p> <p>As far as I can make out, this is the following. Suppose that $G$ is a group and <code>$H&lt;G$</code> a subgroup. (I'm being purposefully vague here about the kinds of groups: finite, Lie,...)</p> <p>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.</p> <p>Then the "permanence relation" seems to say that for every $G$-module $V$ and every $H$-module $W$,</p> <p>$$\operatorname{Ind}(W) \otimes_{\mathbb{C}} V \cong \operatorname{Ind}(W \otimes_{\mathbb{C}}\operatorname{Res}(V))$$ as $G$-modules.</p> <p>This strikes me as the $\otimes$-version of the following isomorphism</p> <p>$$\operatorname{Hom}_H(W,\operatorname{Res}(V)) \cong \operatorname{Hom}_G(\operatorname{Ind}(W),V)$$</p> <p>showing that $\operatorname{Ind}$ and $\operatorname{Res}$ are adjoint functors.</p> <p>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?</p> <p>Thanks in advance.</p> <p><strong>Added</strong></p> <p>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 <a href="http://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula" rel="nofollow">earlier MO question</a> 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!</p>