The ring of conditions of a semisimple group

In this paper we are going to give an explicit description of the so called ring of conditions of an adjoint semisimple algebraic group G, originally introduced in [C. De Concini, C. Procesi, Complete symmetric varieties. II. Intersection theory, in: Algebraic Groups and Related Topics, Kyoto/Nagoya, 1983, in: Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 481–513] in order to study intersection theory in G.