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2 Colloquium Mathematicum

2 2008

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

100%

Dowbor P. , Mróz A.

nr 2

221-282

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 $𝒪((dim_{k} M)⁴)$. 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).

On a separation of orbits in the module variety for domestic canonical algebras

100%

Dowbor P. , Mróz A.

nr 2

283-295

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.