  Polynomials which afford nonnegative, real-rooted symmetric decompositions
have been investigated recently in algebraic, enumerative and geometric
combinatorics. Br\"and\'en and Solus have given sufficient conditions under
which the image of a polynomial under a certain operator associated to
barycentric subdivision has such a decomposition. This paper gives a new proof
of their result which generalizes to subdivision operators in the setting of
uniform triangulations of simplicial complexes, introduced by the first named
author. Sufficient conditions under which these decompositions are also
interlacing are described. Applications yield new classes of polynomials in
geometric combinatorics which afford nonnegative, real-rooted symmetric
decompositions. Some interesting questions in $f$-vector theory arise from this
work.
