The cone of semisimple monoids with the same factorial hull

The factorial hull of the projective variety X (or its cone) is a graded algebra R(X) that can be used in some situations to consider simultaneously all divisor classes on X. Associated with X is a certain cone H in the divisor class group Cl(X) of X. We calculate H explicitly in the case where X is the plongement magnifique for the simple goup G. In this case the elements of H can be represented by semisimple algebraic monoids.