  This paper is concerned with the primitive cohomology of a smooth projective
hypersurface considered as a linear representation for its automorphism group.
Using the Lefschetz-Riemann-Roch formula, the character of this representation
is described on each piece of the Hodge decomposition. A consequence about the
existence of smooth symmetric hypersurfaces that are stable under the standard
irreducible permutation representation of the symmetric group on homogeneous
coordinates is drawn.
