Exceptional modules are tree modules
A representation \(M\) of a finite quiver \(Q\) over a field \(k\) is given by a set \(\{M_x\}\) of finite-dimensional \(k\)-vector spaces indexed by the set \(Q_0\) of vertices of \(Q\) and a set \(\{M_\alpha\}\) of linear transformations \(M_\alpha\colon M_x\to M_y\) for each arrow \(\alpha\colon x\to y\). The sum \(\bigoplus M_x\) can just be seen as a module over the path algebra \(kQ\). Choosing a basis \({\mathcal B}_x\) for each \(x\in Q_0\) and letting \({\mathcal B}=\bigcup{\mathcal B}_x\), it is assigned to the pair \((M,{\mathcal B})\) a quiver \(\Gamma (M,{\mathcal B})\), which is called the coefficient quiver of \(M\) with respect to \(\mathcal B\). The main purpose of the paper under review is to show that if \(M\) is a \(kQ\)-module without self-extensions then there exists a basis \(\mathcal B\) such that \(\Gamma(M,{\mathcal B})\) is a tree, that is, \(M\) is a tree module. It is also shown that such tree modules can be represented by matrices involving as coefficients just 0 and 1.
scientific article; zbMATH DE number 1354726