# Endomorphism ring of trivial source modules for abelian p-groups

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.

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.)