Representation of a quiver with relations (Q1376919)

Given a natural number \(n\) denote by \(Q_n\) the quiver with relations obtained from \(n\) commutative squares by identification of the sink of the \(i\)-th square with the source of the \((i+1)\)-st square for \(i=1,\dots,n-1\). Let \(X\) be a representation of \(Q_n\) over a field \(k\). Given a vertex \(j\) of \(Q_n\) denote by \(X_j\) the space associated to \(j\) in the representation \(X\). A filtration \(0=X^{(m)}\subseteq\cdots\subseteq X^{(1)}\subseteq X^{(0)}=X\) of \(X\) is called admissible provided for any vertices \(i,j\) of \(Q_n\) such that \(i\) is a predecessor of \(j\) in \(Q_n\) the representation \(X\) induces a representation \(X_i\to X_j\) of the one-arrow quiver which decomposes into summands of the form \(X_i^s\to X_j^s\), where \(X^s_l\) is a direct sum complement of \(X^{(s)}_l\) in \(X^{(s-1)}_l\) for \(l=i,j\) and \(s=1,\dots,m\). The main result of the paper asserts that any admissible filtration of \(X\) can be completed to a normal filtration of \(X\) by inserting some new subrepresentations into the chain \(X^{(m)}\subseteq\cdots\subseteq X^{(1)}\subseteq X^{(0)}\). Normality is a property expressing behaviour of certain elements of \(X^{(s)}\) mapped into \(X^{(t)}\) by maps corresponding to arrows in \(Q_n\) for \(s>t\). The result is proved elementarily, the author uses no advanced techniques of representation theory. It is announced in the paper that the result mentioned above is equivalent to a theorem concerning classification of isomorphism types of restrictions of representations of \(Q_n\) to any of the squares forming \(Q_n\). The latter assertion is a special case of a theorem formulated by A. V. Yakovlev in 1975.