An Algorithm related to Compactifications of adjoint Groups

Abstract

A classical problem of projective geometry is the determination of methods for the computation of the number of figures which satisfy certain properties. In the case of subspaces, a method which has been largely developed and is well known is the study of the Grassmann varieties and their Chow ring. In the case of quadrics, pairs of spaces and more in general in the case of a symmetric variety G/H, where G is a semisingle adjoint group, H = G ^σ, σ an order two automorphism of G, a general method has been introduced, defining a suitable intersection ring for G/H. Such ring is difficult to calculate being the direct limit of the Chow ring of all the equivariant regular compactifications of G/H. For a given compactification, one can give a general combinatorial method for the computation of the Chow ring, using root systems, fans and sheaves over fans, which can be treated in a purely algebraic way [BDP].