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Mikhail Ignatyev
Samara State University, Samara, Russia
mihail.ignatev@gmail.com
The orbit method in representation theory
Representation theory is one of the most beautiful and powerful branches of algebra. It has
a lot of applications in geometry, functional analysis, combinatorics, quantum mechanics, etc.
By denition, a representation of a group G is a linear action of G on a vector space V , i.e.,
a map from G to the group of invertible linear operators on V satisfying
(gh) = (g) (h) for all g, h G.
In representation theory, the main problem is as follows: given a group, how one can describe
all its representations? This problem is solved for many important classes of groups: symmetric
groups Sn, Lie groups GLn(C), SLn(C), SOn(C), Sp2n(C), matrix groups over nite elds
GLn(Fq), SLn(Fq), SOn(Fq), Sp2n(Fq), etc.
On the other hand, the main problem of representation theory is still unsolved for many
interesting groups. For example, it is unsolved for the unitriangular group Un, i.e, the group of all
upper-triangular n×n matrices with 1's on the diagonal. In 1962, Alexander Kirillov discovered
wonderful correspondence between representations of this group and certain geometric objects
called coadjoint orbits. This correspondence is called the orbit method. It plays a crucial role in
the description of representations of Un.
I will briey recall some basic facts and results about representations of nite groups. I will
formulate some classical results about representations of groups mentioned above. Then I will
dene coadjoint orbits, explain the orbit method and consider in details the most important
example, the Heisenberg group U3.

Abstract to the talk of M. Ignatyev in JUB-Math-Colloq at 9/9/2013
Abstract to the talk of M. Ignatyev in JUB-Math-Colloq at 9/9/2013


 



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