We consider symmetric (as well as multi-symmetric) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of fixed degrees. We give polynomial (in the dimension of the ambient space) bounds on the number of irreducible representations of the symmetric group which acts on these sets, as well as their multiplicities, appearing in the isotypic decomposition of their cohomology modules with coefficients in a field of characteristic $0$. We also give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets.
Finally, we conjecture that the multiplicities of the irreducible representations of the symmetric group in the cohomology modules of symmetric semi-algebraic sets defined by polynomials of fixed degrees are computable with polynomial complexity, which would imply that the Betti numbers of such sets are also computable with polynomial complexity. This is in contrast with general semi-algebraic sets, for which this problem is provably hard ($\#\mathbf{P}$-hard).
(Joint work with Cordian Riener).