  We present a new, explicit sum formula for symmetric Macdonald polynomials
$P_\lambda$ and show that they can be written as a trace over a product of
(infinite dimensional) matrices. These matrices satisfy the
Zamolodchikov--Faddeev (ZF) algebra. We construct solutions of the ZF algebra
from a rank-reduced version of the Yang--Baxter algebra. As a corollary, we
find that the normalization of the stationary measure of the multi-species
asymmetric exclusion process is a Macdonald polynomial with all variables set
equal to one.
