The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras
Volume 111 / 2008
Abstract
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).
