Can One See the Signs of Structure Constants?

It is described how one can see the signs of action structure constants directly in the weight diagram of microweight and adjoint representations for groups of types E6, E7, and E8. This generalizes the results of the preceding paper, “A third look at weight diagrams”, where a similar algorithm was discussed for microweight representations of E6 and E7. The proofs are purely combinatorial and can be viewed as an elementary construction of Lie algebras and Chevalley groups of types El. In the present paper, which is a sequel of [89, 66, 86], we prove most statements formulated without proofs in §3 of [66] and extend Theorems 1 and 2 of [86] to all microweight modules and to the adjoint modules of types Al, Dl, and El. We give an elementary construction of crystal bases in microweight representations and show how to sight-read the signs of the action structure constants of microweight representations (in a crystal base) and of adjoint representations (in a positive Chevalley base) directly from the weight diagram, alias, from the crystal graph. In particular, this provides elementary purely combinatorial constructions of groups of types E6, E7, and E8 as matrix groups . These constructions could be stated in terms of the graphs depicted in Figures 1–5 in such a manner that any reference to Lie algebras and their representations could be avoided! In [86], similar results and some of their consequences were established for microweight representations of types E6 and E7. There we used realizations of these modules in the unipotent radicals of maximal parabolic subgroups in Chevalley groups of types E7 and E8, respectively. In the present paper we propose straightforward proofs for groups of all types. These proofs are based exclusively upon • identities for the structure constants for Lie algebras, and • geometric and combinatorial properties of microweight representations. Obviously, both for the microweight case and for the adjoint case many different methods are known for computing the signs; these methods are based on one of the following. ◦ Tits inductive algorithm [83, 23, 22, 33, 87]. ◦ Frenkel–Kac cocycle [31, 74, 32, 39, 76, 87]. ◦ Canonical bases of Lusztig–Kashiwara [40, 41], [49]–[51]. ◦ Ringel’s theory of Hall polynomials [71, 89]. ◦ Frenkel–Lepowski cocycle. ◦ Littelmann path model [46]–[48], [56]. 2000 Mathematics Subject Classification. Primary 20G05.