An algorithm for the construction of exceptional modules over tubular canonical algebras.
Let \(\Lambda=kQ/I\) be a finite-dimensional algebra over a field \(k\), defined by a quiver \(Q\) and an admissible ideal \(I\). A basic problem of representation theory is to classify all indecomposable finite-dimensional \(\Lambda\)-modules, up to isomorphism, for a fixed algebra \(\Lambda\) provided \(\Lambda\) is not of wild representation type. The most satisfactory realization of this task would be the construction of a complete list of matrix representations representing all isomorphism classes of indecomposable \(\Lambda\)-modules. By \textit{C. M. Ringel} [Tame algebras and integral quadratic forms. Lect. Notes Math. 1099. Berlin: Springer-Verlag (1984; Zbl 0546.16013)] tubular canonical algebras were introduced and studied. For these algebras, the classification of indecomposables is well known, but not their explicit description by matrices. However for this case all so-called exceptional \(\Lambda\)-modules admit matrix representatives consisting only of \(0\), \(1\) and \(-1\) as coefficients when \(\Lambda\) has only three arms. The main goal of the paper under review is to present a precise recursive algorithm for computing a matrix representative for any isoclass of modules from exceptional tubes over a tubular canonical algebra of quiver type. The input of the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length, and it computes all regular exceptional \(\Lambda\)-modules, or more generally all indecomposable modules in exceptional tubes. The algorithm as well as the collection of ingredient routines are implemented as system of procedures in Maple 9.5 programming language, constituting the main part of the package \texttt{EXCEPTIONAL}.
scientific article; zbMATH DE number 5718177