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 Download: Abstract to the talk of M. Ignatyev in JUB-Math-Colloq at 9/9/2013 (application/pdf 45.9 KB)

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