  We investigate the arithmetic formula complexity of the elementary symmetric
polynomials S(k,n). We show that every multilinear homogeneous formula
computing S(k,n) has size at least k^(Omega(log k))n, and that product-depth d
multilinear homogeneous formulas for S(k,n) have size at least
2^(Omega(k^{1/d}))n. Since S(n,2n) has a multilinear formula of size O(n^2), we
obtain a superpolynomial separation between multilinear and multilinear
homogeneous formulas. We also show that S(k,n) can be computed by homogeneous
formulas of size k^(O(log k))n, answering a question of Nisan and Wigderson.
Finally, we present a superpolynomial separation between monotone and
non-monotone formulas in the noncommutative setting, answering a question of
Nisan.
