  For Majorana-Wilson lattice fermions in two dimensions we derive a dimer
representation. This is equivalent to Gattringer's loop representation, but is
made exact here on the torus. A subsequent dual mapping leads to yet another
representation in which a highly efficient Swendsen-Wang type cluster algorithm
is constructed. It includes the possibility of fluctuating boundary conditions.
It also allows for improved estimators and makes interesting new observables
accessible to Monte Carlo. The algorithm is compatible with the Gross-Neveu as
well as an additional Z(2) gauge interaction. In this article numerical
demonstrations are reported for critical free fermions.
