Geometry of standard symmetrized tensors

The geometric properties of the set of standard (decomposable) symmetrized tensors are studied and some general results are obtained. As an example, the geometry is worked out completely in the case where the group is a dihedral group, and this result is used to give a more conceptual proof of an earlier result. As another example, it is shown that there exists an orbital subspace such that the standard symmetrized tensors in the subspace form a root system isomorphic to a given irreducible root system if and only if the irreducible root system is simply laced.