Yakovleva, A. A. Representation of a quiver with relations. (English. Russian original) Zbl 0902.16014 J. Math. Sci., New York 89, No. 2, 1172-1179 (1998); translation from Zap. Nauchn. Semin. POMI 227, 140-151 (1995). 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. Reviewer: S.Kasjan (Toruń) Cited in 1 Review MSC: 16G20 Representations of quivers and partially ordered sets 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) Keywords:representations of quivers with relations; admissible filtrations; normal filtrations; restrictions of representations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] A. V. Yakovlev, ”Indecomposable representations of oriented graphs of certain type”, in:Soviet Algebraic Symposium. Abstract, Gomel (1975), pp. 473–474. [2] L. A. Nazarova, ”Representations of a quadruple”,Izv. Akad. Nauk SSSR, Ser. Mat.,31, 1361–1377 (1967). [3] A. V. Yakovlev, ”2-adic integral representations of a cyclic group of order 8”,Zap. Nauchn. Semin. LOMI,28, 98–123 (1972). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.