Constructing Homomorphism Spaces and Endomorphism Rings

Abstract We present a new deterministic algorithm for constructing homomorphism spaces and endomorphism rings of finite dimensional modules. The modules are given via vertex projective presentations over path algebras and finite dimensional quotients of path algebras. We use the theory of right Gröbner bases to encode modules and to construct appropriate systems of equations for computing homomorphism spaces and endomorphism rings. The algorithm is implemented in the computer algebra system GAP and is included in Hopf, a computational package for noncommutative algebra. The performance of our implementation for computing endomorphism rings is experimentally compared with the implementation in Magma for the same class of modules. These experiments show that our implementation has a better time complexity.