Subspace quiver ($S_r$)

This quiver has $r$ source vertices, each with one arrow to a common sink. Its path algebra has the vertex idempotents and the arrows as basis. After extracting source kernels from a representation, its remaining arrows identify the source spaces with subspaces of the sink.

The four-subspace quiver has four sources with arrows into one sink. Four one-dimensional sources mapped to the lines $k{e}_1$, $k{e}_2$, $k(e_1+e_2)$ and $k(\lambda e_1+e_2)$ in $k^2$ give a brick module for every $\lambda$. Preserving the first three lines forces scalar endomorphisms. Its Tits form of a quiver is zero, so the Ringel form gives one-dimensional self-extensions.

The subspace quiver with three sources reduces to the problem of three subspaces in one vector space. The three-subspace decomposition gives eleven indecomposables with vertex dimensions at most one, and one exceptional indecomposable of the three-subspace quiver.

Every triple of subspaces is a direct sum of one-dimensional membership blocks and two-dimensional three-line blocks. Split off the triple intersection, then the pairwise intersections using projections killing the third subspace. Next split complements to $U_i\cap (U_j+U_{\ell })$, using projections killing the other two subspaces. The remaining subspaces are pairwise disjoint and each lies in the sum of the other two. Their sum is $U_1\oplus U_2$, with $U_3$ the graph of an isomorphism $U_1\to U_2$. A graph basis splits this into two-dimensional blocks. A complement to the total span supplies zero-membership blocks. This elementary argument classifies the three-subspace quiver.

Take sources $k,k,k$, sink $k^2$, and arrows sending $1$ to $e_1,e_2,e_1+e_2$. An endomorphism preserving these three lines must be scalar, so the representation has endomorphism ring $k$ and is a brick module. It is the only indecomposable of the three-subspace quiver with a two-dimensional sink.