  For a polynomial ring S in n variables, we consider the natural action of the
symmetric group S_n on S by permuting the variables. For an S_n-invariant
monomial ideal I in S and j >= 0, we give an explicit recipe for computing the
modules Ext^j(S/I,S), and use this to describe the projective dimension and
regularity of I. We classify the S_n-invariant monomial ideals that have a
linear free resolution, and also characterize those which are Cohen-Macaulay.
We then consider two settings for analyzing the asymptotic behavior of
regularity: one where we look at powers of a fixed ideal I, and another where
we vary the dimension of the ambient polynomial ring and examine the invariant
monomial ideals induced by I. In the first case we determine the asymptotic
regularity for those ideals I that are generated by the S_n-orbit of a single
monomial by solving an integer linear optimization problem. In the second case
we describe the behavior of regularity for any I, recovering a recent result of
Murai.
