Last week David taught us a new set of generators of the Chow ring of a matroid called the simplicial generators. In the special case when our matroid is the uniform matroid of rank and dimension n, the Chow ring agrees with the Chow ring of the permutahedron, a particularly nice toric variety. I will show that in this case, the simplicial generators correspond to nested quotients of matroids on n elements. The tool I will introduce relating the two are Minkowski weights, which give an alternate description of the Chow rings of toric varieties. Either at the end of my talk or in our talk week, we will use this relationship to analyze the kernel of the surjection from the Chow ring of the permutahedron \Sigma_n to the Chow ring of a matroid with ground set [n], leading to a proof of Poincare duality.