Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)?
Since $KG$ is local, the quiver should easily be obtainable via the radical $J$ as $J/J^2$. But I do not know how to obtain the relations and whether this can be done in a quick way with a computer.
Perhaps the QPA function AlgebraAsQuiverAlgebra can be used to obtain what you want?
I tested the following small example:
LoadPackage("qpa");
p := 3;
K := GF(p);
G := CyclicGroup(p);
KG := GroupRing(K, G);
A := AlgebraAsQuiverAlgebra(KG)[1];
R := RelationsOfAlgebra(A);
AlgebraAsQuiverAlgebra instead of IsomorphismFpAlgebra. Can you check if the result is now correct?
$\endgroup$
Here is a computation of the quiver. I don't know how to get the relations. Let $G$ be a finite $p$-group and $K$ the $p$-element field. Note that the trivial module is the unique simple module and the radical $J$ is the augmentation ideal. It is well known that $J$ has basis the elements of the form $g-1$ with $g\in G\setminus\{1\}$. Note that
$(g-1)(h-1)= gh-1-(g-1)-(h-1)$
and so
$gh-1+J^2=(g-1)+J^2+ (h-1)+J^2$.
Thus $J/J^2$ is a $K$-vector space with generators $[g]$ with $g\in G$ and relations $[g]+[h]=[gh]$. In other words, $J/J^2$ is the universal elementary abelian $p$-group image of $G$, which is $G/\Phi(G)$ with $\Phi(G)$ the Frattini subgroup of $G$ (the intersection of all maximal subgroups $=[G,G]G^p$).
Thus $\dim J/J^2 = \log_p |G/\Phi(G)|$ and so you have one vertex and that number of loops. I would guess GAP could find a set of generators for the Frattini quotient and then your path algebra would map the loop corresponding to $x\Phi(G)$ to $x-1$ and I assume there are algorithms to get the kernel.
