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GL_q(N)- and SO_q(N)-covariant deformations of the completelysymmetric/antisymmetric projectors with an arbitrary number of indices areexplicitly constructed as polynomials in the braid matrices. The preciserelation between the completely antisymmetric projectors and the completelyantisymmetric tensor is determined. Adopting the GL_q(N)- and SO_q(N)-covariantdifferential calculi on the corresponding quantum group covariantnoncommutative spaces C_q^N, R_q^N, we introduce a generalized notion ofvielbein basis (or "frame"), based on differential-operator-valued 1-forms. Wethen give a thorough definition of a SO_q(N)-covariant R_q^N-bilinear Hodge mapacting on the bimodule of differential forms on R_q^N, introduce the exteriorcoderivative and show that the Laplacian acts on differential forms exactly asin the undeformed case, namely it acts on each component as it does onfunctions.Comment: latex file, 24 pages. Some citations added and misprints corrected.  Final version to appear in J. Phys. A Math. and Ge

Quantum group covariant (anti)symmetrizers, -tensors, vielbein, Hodge map and Laplacian