Functors on categories of finite group representations
In this interesting paper the author studies induction and restriction on the functor category from finitely generate \(\bar kG\)-modules to \(\bar k- \)spaces, \(\bar k\) a field and G a finite group. This is of great importance from the representation theory of G because of the work of Auslander, Reiten e.a. The functors ''induction'', ''restriction'' and ''conjugation'' are defined for functors on the level of modules. These functors enjoy nice properties, e.g. Mackey's formula holds. In connection with relative projectivity of finitely presented functors the existence of vertices and sources is shown - also here the vertex is a p-group, \(p=char \bar k\). Many of the results for modules hold for functors; however, a projective functor is not necessarily \(\{\) \(1\}\)- projective. Using this and a relative trace map, the author obtains an elegant proof of Carlson's strong correspondence theorem. To mention just one of the many interesting results in this paper: Theorem: Let V be an indecomposable \(\bar kG\)-module with vertex P and source S. Let H be a subgroup of G which contains the stabilizer of S in G. Let W be an indecomposable summand of \(V_ H\) which has vertex P and source S. If \(0\to W_ 2\to W_ 1\to W\to 0\) is an almost split sequence of \(\bar kH\)-modules, then \(0\to W_ 2\uparrow^ G\to W_ 1\uparrow^ G\to W\uparrow^ G\to 0\) is the direct sum of the almost split sequence \(0\to V_ 2\to V_ 1\to V\to 0\) and a split exact sequence.
scientific article; zbMATH DE number 3904811
scientific article