  It is well-known that the convex and concave envelope of a multilinear
polynomial over a box are polyhedral functions. Exponential-sized extended and
projected formulations for these envelopes are also known. We consider the
convexification question for multilinear polynomials that are symmetric with
respect to permutation of variables. Such a permutation-invariance structure
naturally implies a quadratic-sized extended formulation for the envelopes
through the use of disjunctive programming. The optimization and separation
problems are answered directly without using this extension. The problem
symmetry also implies that permuting the coefficients of a core set of facets
generates all the facets. We provide some necessary conditions for a valid
inequality to be a core facet. These conditions are applied to obtain envelopes
for two classes, symmetric supermodular functions and multilinear monomials
with reflection symmetry, thereby yielding alternate proofs to literature.
Furthermore, we use constructs from the reformulation-linearization-technique
to completely characterize the points lying on each facet.
