ABSTRACT
It is shown that the galleries configurations scheme C onf G ( g ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315367309/5389696d-8ee9-4237-ac8b-c1bdba9ca57c/content/eq6500.tif"/> of a split reductive S-group scheme G, defined by a fixed gallery of types g, is isomorphic to a Contracted Product. This isomorphism gives rise to natural parametrizations of the C onf G ( g ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315367309/5389696d-8ee9-4237-ac8b-c1bdba9ca57c/content/eq6501.tif"/> -Cells. By means of a parametrization it is proved that there is a Cell of C onf G ( g ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315367309/5389696d-8ee9-4237-ac8b-c1bdba9ca57c/content/eq6502.tif"/> which is an open relatively schematically dense subscheme (The Big Open Cell). A Contracted Product may be decomposed in a sequence of locally trivial fibrations with typical fiber G/P, where P is a parabolic subgroup of G. The Big Open Cell is isomorphic to a contracted product of big open cells of homogeneous spaces G/P. By means of the Contracted Product the image of a minimal gallery type configuration is calculated by the Retraction on an apartment. This calculation amounts to determining the fibers of the resolving morphism.
