Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 3 / 3 / b

The four-subspace quiver has four one-dimensional sources and a two-dimensional sink. Its Tits form of a quiver at this dimension vector is $q=4\cdot 1^2+2^2-4(1\cdot 2)=0$.

An endomorphism comprises source scalars $a_1,\dots ,a_4$ and a sink matrix $T$. The first two columns $e_1,e_2$ force $T=\mathrm{diag}(a_1,a_2)$. The third column $e_1+e_2$ then forces $a_1=a_2=a_3$, making $T$ scalar. Since the fourth column $(\lambda ,1{)}^T$ is always nonzero, its scalar is also the same, for every $\lambda$.

Thus ${\mathrm{End}}_Q(V)=k$, so the representation is a brick module. The Ringel form givesThis conclusion also covers $\lambda =0,1$, where the fourth line repeats one of the earlier lines; the first three lines already force scalar endomorphisms.