The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach
Volume 107 / 2007
Colloquium Mathematicum 107 (2007), 221-261
MSC: 16G60, 16G70, 68Q99.
DOI: 10.4064/cm107-2-4
Abstract
Given a module $M$ over an algebra ${\mit\Lambda}$ and a complete set ${\cal{X}}$ of pairwise nonisomorphic indecomposable ${\mit\Lambda}$-modules, the problem of determining the vector $ m(M)=(m_X)_{X\in {\cal{X}}}\in {\mathbb N}^{\cal{X}}$ such that $M\cong \bigoplus_{X\in \cal {X}}X^{m_X}$ is studied. A general method of finding the vectors $ m(M)$ is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type $\widetilde{\mathbb{A}}_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).
