  We study symmetric nonnegative forms and their relationship with symmetric
sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric
nonnegative forms and symmetric sums of squares form closed, convex cones in
the vector space of $n$-variate symmetric forms of degree $2d$. Using
representation theory of the symmetric group we characterize both cones in a
uniform way. Further, we investigate the asymptotic behavior when the degree
$2d$ is fixed and the number of variables $n$ grows. Here, we show that, in
sharp contrast to the general case, the difference between symmetric
nonnegative forms and sums of squares does not grow arbitrarily large for any
fixed degree $2d$. We consider the case of symmetric quartic forms in more
detail and give a complete characterization of quartic symmetric sums of
squares. Furthermore, we show that in degree $4$ the cones of nonnegative
symmetric forms and symmetric sums of squares approach the same limit, thus
these two cones asymptotically become closer as the number of variables grows.
We conjecture that this is true in arbitrary degree $2d$.
