A CLOSER LOOK AT WEIGHT DIAGRAMS

The talk presents recent enhancements of the
representation theoretic and geometric methods
in the study of exceptional groups. In the
lectures at TIFR in 2011, and in the previous
lectures this week in the framework 
ATM Workshop on Classical and Non-stable Algebraic K-Theory
I sketched the use of minimal modules
for calculations in exceptional groups over
rings. Essentially, I explained, that weight
diagrams are an efficient substitute of matrices,
when you deal with an individual column or row
of a matrix. However, applications at the level
of $K_2$ and more sophisticated applications at
the level of $K_1$ require more. I outline some
new results, which indicate that with the use
of weight diagrams you can do much more, and
easily calculate with several columns and/or
rows of elements of exceptional groups. I mention
several current or potential applications, such
as stability for $K_1$ and $K_2$, description of
subnormal subgroups, overgroups of subsystem
subgroups, structure of not necessarily split
isotropic groups, etc.