Affiliations: Faculty of Mathematics and Computer Science, Nicolaus
Copernicus University, Chopina 12/18, 87-100 Toruń, Poland.
E-mail: [email protected]
Abstract: We give an algorithmic solution in a simple combinatorial data of
Birkhoff0s type problem studied in [22] and [25], for the category
rep_{ft}(I,K[t]/(t^m)) of filtered I-chains of modules over the K-algebra K[t]/(t^m)
of K-dimension m <∞, where m ⩾ 2, I is a finite poset with a unique maximal
element, and K is an algebraically closed field. The problem is to decide when
the indecomposable objects of the category rep_{ft}(I,K[t]/(t^m)) admit a
classification by means of a suitable parametrisation. A complete solution of
this important problem of the modern representation theory is contained in
Theorems 2.4 and 2.5. We show that rep_{ft}(I,K[t]/(t^m)) admits such a
classification if and only if (I,m) is one of the pairs of the finite list
presented in Theorem 2.4, and such a classification does not exist for
rep_{ft}(I,K[t]/(t^m)) if and only if the pair (I,m) is bigger than or equal to one
of the minimal pairs of the finite list presented in Theorem 2.5. The finite lists are constructed by producing computer accessible algorithms and
computational programs written in MAPLE and involving essentially the package
CREP (see Section 4). On this way the lists are obtained as an effect of
computer computations. In particular, the solution we get shows an importance
of the computer algebra technique and computer computations in solving
difficult and important problems of modern algebra.
Keywords: poset, indecomposable linear representations, Birkhoff type problem, chains of vector spaces with linear operators, tame representation type, classification problems
