I am going to investigate separating invariants and just invariants of
exceptional simple groups acting on several copies of their fundamental
representations (algebra of octonions, Albert algebra etc.), over an infinite
field of arbitrary characteristic.
More precisely, let G be an exceptional simple algebraic group of type F_4 or
E_6. This group acts on the 27-dimensional simple Jordan algebra A (Albert
algebra), hence on several copies of A, diagonally. The invariants of this
action are known only over a field of zero characteristic. The case of
positive characterist (modular case) is still open and extremely difficult
problem.
If the characteristic of the ground field is sufficiently large ( > 13), then
the calculating of arbitrary invariants of the above action can be reduced
to the calculating of multi-linear ones. But even the last problem is very
difficult, and
this is the first problem that I am going to attack (with the help of RA).
In many cases when the direct computing of invariants is difficult, one can
try to describe separating invariants, i.e. those invariants are separating
closed orbits of the given action. Most fundamental property of separating
invariants is that the algebra of just invariants is a p-closure of the
subalgebra generated by separating invariants. So, I am going to find minimal
sets of separating invariants of the diagonal action of G on several copies of
Albert algebra.
Minimum Qualification
I am going to investigate separating invariants and just invariants of
exceptional simple groups acting on several copies of their fundamental
representations (algebra of octonions, Albert algebra etc.), over an infinite
field of arbitrary characteristic.
Regarding this, a candidate for RA position, which is supposed to assist me in
realizing this project, should be aware of :
1) Fundamental facts and techniques in the algebraic group theory (both
structural and representation).
To be familiar with the book J.E.Humphreys "Linear Algebraic Groups" is the
absolutely necessary minimum to satisfy these requirements.
2) Fundamental facts and techniques in the invariant theory (invariants of
rational action of algebraich groups acting on affine varieties/schemes,
semi-invariants and separating invariants).
To be familiar with the book H.Derksen, G.Kemper "Computational Invariant
Theory" the absolutely necessary minimum to satisfy these requirements.
Preferred Qualification
The candidate should have at least two articles in the invariant theory of
algebraic groups.
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