The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor¹ ; Andrzej Mróz¹

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

Colloquium Mathematicum, Tome 111 (2008) no. 2, pp. 221-282

Résumé

Given a module $M$ over a domestic canonical algebra ${\mit\Lambda}$ and a classifying set $ \boldsymbol{X}$ for the indecomposable ${\mit\Lambda}$-modules, the problem of determining the vector $ m(M)=(m_x)_{x\in {\boldsymbol{X}}}\in {\mathbb N}^{\boldsymbol{X}}$ such that $M\cong \bigoplus_{x\in \boldsymbol{X}} X_x^{m_x}$ is studied. A precise formula for $\mathop{\rm dim}\nolimits_k\mathop{\rm Hom}_{\mit\Lambda}(M,X)$, for any postprojective indecomposable module $X$, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors $ m(M)$ presented by the authors in Colloq. Math. 107 (2007) leads to algorithms with the complexity ${\cal O}((\mathop{\rm dim}\nolimits_k M)^4)$. A precise description of algorithms determining the multiplicities $m(M)_x$ for postprojective roots $x\in \boldsymbol{X}$ is given (Algorithms 6.1, 6.2 and 6.3).

Détail

DOI : 10.4064/cm111-2-6

Keywords: given module domestic canonical algebra mit lambda classifying set boldsymbol indecomposable mit lambda modules problem determining vector boldsymbol mathbb boldsymbol cong bigoplus boldsymbol studied precise formula mathop dim nolimits mathop hom mit lambda postprojective indecomposable module computed theorem interrelations between various structures set postprojective roots described theorem proved theorem general method finding vectors presented authors colloq math leads algorithms complexity cal mathop dim nolimits precise description algorithms determining multiplicities postprojective roots boldsymbol given algorithms nbsp nbsp nbsp