On linear shift representations

We introduce and develop the concept of (linear) shift representation. This derives from a certain action on 2-cocycle groups that preserves both cohomological equivalence and orthogonality for cocyclic designs, discovered by K.J. Horadam. Detailed information about fixed point spaces and reducibility is given. We also discuss results of computational experiments, including the calculation of shift orbit structure and searching for orthogonal cocycles.