Computing Homomorphism Spaces between Modules over Finite Dimensional Algebras

In this paper, we study the fundamental algorithmic problem of determining the homomorphisms from an Amodule M to an A-module N over an algebra A. We shall outline an algorithm for the case that A is a finite dimensional algebra over a finite field F and both modules M and N are finitely generated as A-modules. More specifically, the input to our algorithm consists of the following data: The modulesM and N are given in terms of linear algebra via the matrices for a generating system of the F -algebra A. Note that this is a common situation in the study of representations for a given finite dimensional algebra. The output of the algorithm will be matrices that form an F -basis for the F -vector space of A-homomorphisms from M to N . For a different set-up in the case that A is commutative, see the end of the introduction. In caseM = N , we have determined an F -basis of the endomorphism ring ofM . One of the reasons for focusing on the endomorphism ring of a module is given by the fact that we can use the knowledge about the structure of the endomorphism ring to give a decomposition ofM into indecomposable A-modules. In a separate paper, we shall give an algorithm that determines such a decomposition of M (see [Lux and Szőke 03]). As a starting point of our investigation of a homomorphism space from an A-module M to an A-module N , we make use of well-known algorithms that can answer structural questions about a given module. For example, the composition factors, the socle, and the radical series can be determined algorithmically; see [Parker 84], [Lux