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An Algorithmic Solution of a Birkhoff Type Problem

Simson D., Wojewódzki M.

Fundamenta Informaticae

2008

Vol. 83, nr 4

389-410

We give an algorithmic solution in a simple combinatorial data of Birkhoff?s type problem studied in [22] and [25], for the category repft(I, K[t]/(tm)) of filtered I-chains of modules over the K-algebra K[t]/(tm) of K-dimension m < ?, where m ^(3) 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 repft(I, K[t]/(tm)) 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 repft(I, K[t]/(tm)) 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 repft(I, K[t]/(tm)) 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.

On stabilizers of G-atoms of representation-tame categories

Dowbor P.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

303-315

Given a locally bounded representation-tame k-category R over an algebraically closed field k and a torsionfree group G of k-linear automorphims of R the stabilizers of infinite G-atoms over R are described (Theorem 2.1). Representation embeddings of the module category over n-flowers induced by indecomposable locally finite dimensional modules are studied (Theorem 2.3)