While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum. 


Here is a special case. Let $G$ be a group, with $H$ a subgroup of index $2$. Choose $s \in G$ which is not in $H$. For $(\pi,V)$ a representation of $H$, define a representation $\sigma$ of $G$ with underlying space $V \otimes V$ as follows: if $h \in H$, define $\sigma(h) = \pi(h) \otimes \pi(shs^{-1})$. If $g \in G, \not\in H$, define $\sigma(g)$ on generators by


$$\sigma(g) v \otimes w = \pi(gs^{-1})w \otimes \pi(sg)v$$


If we had used $V \oplus V$ instead of $V \otimes V$, then following the above construction would have produced the representation $\operatorname{Ind}\_H^G(\pi)$. So this is like a tensor product version of induced representation.


I found some papers which talk about "tensor induction," and I believe the above construction is a very simple case of that. What is tensor induction, generally speaking? Is it adjoint to some functor, or satisfy a universal property? 



Reference request: tensor induction
An introduction to tensor induction is given in §13 of


*Curtis, Charles W.; Reiner, Irving*, Methods of representation theory with applications to finite groups and orders. Volume 1., Wiley Classics Library; New York et al.: John Wiley & Sons. (1990). [ZBL0698.20001](https://zbmath.org/?q=an:0698.20001). 


Some papers also contain brief introductions to tensor induction, for example:


*Isaacs, I. M.*, [**Character correspondences in solvable groups**](http://dx.doi.org/10.1016/0001-8708(82)90037-8), Adv. Math. 43, 284-306 (1982). [ZBL0487.20004](https://zbmath.org/?q=an:0487.20004).  

(Both Curtis-Reiner and Isaacs have more references on tensor induction.)


A universal property of tensor induction is given in my paper (Thm. 2.1):


*Ladisch, Frieder*, [**Corestriction for algebras with group action.**](http://dx.doi.org/10.1016/j.jalgebra.2015.06.015), J. Algebra 439, 438-453 (2015). [ZBL1330.16026](https://zbmath.org/?q=an:1330.16026). [arXiv:1409.3166](http://www.arxiv.org/abs/1409.3166). 


The universal property is in terms of the natural multilinear map from the induced module into the tensor induced module. 


As tensor induction is not an additive functor, it can not be a (left or right) adjoint of a functor. In particular, there is no analog of Frobenius reciprocity.


