A Third Look at Weight Diagrams

In this paper, which is a sequel of PSV], we develop a completely elementary approach to calculations in Chevalley groups G = G((; R) of types = E 6 and E 7 over a commutative ring R using only the weight diagrams (alias, crystal graphs) of their minimal modules. After an elementary construction of a crystal base we explicitly describe action of root subgroups and of the extended Weyl group, multilinear invariants, equations deening the orbit of the highest weight vector and Freudenthal tranvections. As an illustration of our methods we give the rst complete a priori proof of the central step in the method of decomposition of unipotents (see VPS], V2], VP], SV], VPe]) for these cases. Namely we prove that any singular column v is stabilised by a non-trivial Freudenthal transvection of a certain type (`fake root unipotent') and that there are in fact enough of those to generate the whole elementary group of type over R as the v ranges over the columns of a matrix g 2 G. It is known that this result immediately implies the main structure theorems for G (description of normal subgroups, standard commutator formulae and the like). The results of the present paper provide complete proofs for the algebraic part of V2] in the cases of E 6 and E 7 , complete proofs for the geometric part of the above paper are given in V7].