An Algorithmic Solution of a Birkhoff Type Problem

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.