Let R be a reduced root system in a finite dimensional vector space V, N the associated weight lattice, and &phis; the fan of Weyl chambers in N. The pair (N, &phis;) determines a smooth, projective toric variety X = X(R). The action of the Weyl group W on N induces an action of W on X and thus an action on the integral and rational cohomology of X. In work of J. Stembridge, I. Dolgachev and V. Lunts it is shown that in the An and Cn cases, the Weyl group action on the cohomology is a graded permutation representation. An interesting problem, suggested in a paper by J. Stembridge, is to find a concrete basis for this cohomology permuted by W . In this thesis we give a geometric solution to this problem. An immediate consequence is a concrete, and geometrically natural formula for the isotypic Betti numbers of the X(An) cohomology. The key idea is the introduction of natural projection and inclusion maps, which allow us to naturally decompose the cohomology of X( An) and X(Cn). This decomposition allows us to further give an integral basis for the cohomology of X(An) and X(Cn). We also study the X(Bn) variety, describing it geometrically, giving a natural proof of the equivalence of the module structure of its cohomology with the module structure of the cohomology of X( Cn), and giving a proof that for n > 2, the varieties X(Bn) and X(Cn) are not isomorphic.
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