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Bernhard Böhmler  (who is also on MO) and myself had the following idea: Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides the order of $G$.

Let $A=kG$ be the group algebra of $G$ and $M$ the direct sum of indecomposable all trivial source modules (that are modules which are indecomposable direct summands of modules of the form ${k\!\uparrow}_H^G$ for some $p$-subgroup $H$ of $G$).

One might ask what properties $B:=End_{kG}(M)$ has.

Quesion 1: Is $B$ studied already in the literature?

The simplest case is when $G$ is abelian and then we can also assume that $G$ is an abelian $p$-group. Then any indecomposable direct summand of $M$ is of the form $k(G/H_i)$ for some subgroup $H_i$ of $G$.

Question 2: When $G$ is an (elementary) abelian $p$-Group, is $B$ a Gorenstein ring?

It might also be interesting whether the relations of $B$ have an easy description, since the Hom-spaces can in principle be described purely combinatorially. We can show that $B$ has dominant dimension equal to $2$.

Our question has a positive answer when $G$ is cyclic and then $B$ has Gorenstein dimension $2$. When $G$ is the Klein four group it is also true and $B$ has Gorenstein dimension 3. One can show that the quiver of $B$ is given by doubling the Hasse quiver of the poset of subgroups of $G$ (that is for every arrow in the Hasse quiver we add the opposite arrow).

For non-abelian groups it is not true, the quaternion group gives a counterexample.

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Representations of $B$ (or at least an equivalent category) are studied in the literature under the name of "cohomological Mackey functors".

Theorem 1.1 of

Bouc, Serge; Stancu, Radu; Webb, Peter, On the projective dimensions of Mackey functors, Algebr. Represent. Theory 20, No. 6, 1467-1481 (2017). ZBL1422.20005

implies that $B$ is Gorenstein if and only if the Sylow $p$-subgroups of $G$ are cyclic or dihedral. (In the latter case $p$ must be $2$, of course.)

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