  We study nonnegative and sums of squares symmetric (and even symmetric)
functions of fixed degree. We can think of these as limit cones of symmetric
nonnegative polynomials and symmetric sums of squares of fixed degree as the
number of variables goes to infinity. We compare these cones, including finding
explicit examples of nonnegative polynomials which are not sums of squares for
any sufficiently large number of variables, and compute the tropicalizations of
their dual cones in the even symmetric case. We find that the tropicalization
of the dual cones is naturally understood in terms of the overlooked
superdominance order on partitions. The power sum symmetric functions obey this
same partial order (analogously to how term-normalized power sums obey the
dominance order).
