Effective Wedderburn and applications

We propose a new algorithmic proof of the classical Wedderburn’s theorem in the representation theory of finite abelian groups. Namely, the algebra isomorphisms, known to exist from the classical theory, are explicitly found. This leads to a new succinct resolution of the well-known isomorphism problem for group rings and an alternative description of the automorphisms of these algebras. Employing ideas from algebraic geometry, the global variety of all representations of fixed dimension for a finite group is introduced and its geometric properties are explored. The last step involves an extensive computer assisted analysis of the resulting high dimensional polynomial ideals.