The cohomology groups (with rational coefficients) of semi-algebraic sets defined by symmetric polynomials inherit a structure of a finite dimensional module over the symmetric group (from the action of the symmetric group on the ambient space). The isotypic decomposition of these modules shed important information on the Betti numbers of such sets, via the multiplicities of the various irreducible representations (the so called Specht modules), and the well known ``hook formula'' that gives the dimensions of these irreducible representations. We prove new vanishing results on the multiplicities of these Specht modules in the cohomology groups of semi-algebraic sets defined by symmetric polynomials (in each dimension).