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We consider symmetric (under the action of products of finite symmetricgroups) real algebraic varieties and semi-algebraic sets, as well as symmetriccomplex varieties in affine and projective spaces, defined by polynomials ofdegrees bounded by a fixed constant $d$. We prove that if a Specht module,$\mathbb{S}^\lambda$, appears with positive multiplicity in the isotypicdecomposition of the cohomology modules of such sets, then the rank of thepartition $\lambda$ is bounded by $O(d)$. This implies a polynomial (in thedimension of the ambient space) bound on the number of such modules.Furthermore, we prove a polynomial bound on the multiplicities of those that doappear with positive multiplicity in the isotypic decomposition of the abovementioned cohomology modules.  We give some applications of our methods in proving lower bounds on thedegrees of defining polynomials of certain symmetric semi-algebraic sets, aswell as improved bounds on the Betti numbers of the images under projections of(not necessarily symmetric) bounded real algebraic sets, improving in certainsituations prior results of Gabrielov, Vorobjov and Zell.

On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities