Two of the most important and fundamental results in the representation theory of a reductive algebraic group $\mathbf{G}$ (over an algebraically closed field of positive characteristic) are the linkage principle and the translation principle.

The linkage principle asserts that the category $\mathrm{Rep}(\mathbf{G})$ of finite dimensional $\mathbf{G}$-modules decomposes into subcategories called linkage classes, and the translation principle relates the different linkage classes via so-called translation functors.

Combining these two results, one sees that many problems in the representation theory of $\mathbf{G}$ can be reduced to questions about a single linkage class.

However, this strategy fails for two reasons when one tries to study tensor products of $\mathbf{G}$-modules.

Firstly, the linkage classes are not closed under tensor products, and secondly, it is a priori not clear how structural information about tensor products of $\mathbf{G}$-modules in a fixed linkage class can be used to deduce (precise) structural information about tensor products of $\mathbf{G}$-modules in arbitrary linkage classes.

In this talk, I will explain how one can (partially) overcome these obstacles using a `translation principle for tensor products'.