The Weitzenb\"ock Machine

In this article we give a unified treatment of the construction of all possible Weitzenb\"ock formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the space of Weitzenb\"ock formulas on the one hand and characterize Weitzenb\"ock formulas as eigenvectors for an explicitly known matrix on the other. Both classifications allow us to find tailor--suit Weitzenb\"ock formulas for applications like eigenvalue estimates or Betti number estimates.