The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor ¹   ; Andrzej Mróz ¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 107 (2007) no. 2, pp. 221-261

Résumé

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).

Détail

DOI : 10.4064/cm107-2-4

Keywords: given module algebra mit lambda complete set cal pairwise nonisomorphic indecomposable mit lambda modules problem determining vector cal mathbb cal cong bigoplus cal studied general method finding vectors presented corollary theorem corollary discussed applied practice classes algebras string algebras finite representation type hereditary algebras type widetilde mathbb second detailed algorithms given algorithms