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

Piotr Dowbor, and Andrzej Mróz. "The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras." Colloquium Mathematicae 111.2 (2008): 221-282. <http://eudml.org/doc/286364>.

@article{PiotrDowbor2008,
abstract = {Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m(M) = (m_\{x\})_\{x∈X\} ∈ ℕ^\{X\}$ such that $M ≅ ⨁_\{x∈X\} X_\{x\}^\{m_\{x\}\}$ is studied. A precise formula for $dim_\{k\} Hom_\{Λ\}(M,X)$, 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 $((dim_\{k\} M)⁴)$. A precise description of algorithms determining the multiplicities $m(M)_\{x\}$ for postprojective roots x ∈ X is given (Algorithms 6.1, 6.2 and 6.3).},
author = {Piotr Dowbor, Andrzej Mróz},
journal = {Colloquium Mathematicae},
keywords = {domestic canonical algebras; multiplicity vectors; postprojective indecomposable modules; postprojective roots; algorithms; multiplicities; representations; decompositions},
language = {eng},
number = {2},
pages = {221-282},
title = {The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras},
url = {http://eudml.org/doc/286364},
volume = {111},
year = {2008},
}

TY - JOUR
AU - Piotr Dowbor
AU - Andrzej Mróz
TI - The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras
JO - Colloquium Mathematicae
PY - 2008
VL - 111
IS - 2
SP - 221
EP - 282
AB - Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m(M) = (m_{x})_{x∈X} ∈ ℕ^{X}$ such that $M ≅ ⨁_{x∈X} X_{x}^{m_{x}}$ is studied. A precise formula for $dim_{k} Hom_{Λ}(M,X)$, 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 $((dim_{k} M)⁴)$. A precise description of algorithms determining the multiplicities $m(M)_{x}$ for postprojective roots x ∈ X is given (Algorithms 6.1, 6.2 and 6.3).
LA - eng
KW - domestic canonical algebras; multiplicity vectors; postprojective indecomposable modules; postprojective roots; algorithms; multiplicities; representations; decompositions
UR - http://eudml.org/doc/286364
ER -