Mathematics Colloquium

Mikhail Ignatyev

(Samara State University)

"The orbit method in representation theory"

Date: Mon, September 9, 2013
Time: 17:15
Place: Research II Lecture Hall

Abstract: 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 definition, a \emph{representation} of a group $G$ is a linear action of $G$ on a vector space $V$, i.e., a~map $\phi$ from $G$ to the group of invertible linear operators on $V$ satisfying $$\phi(gh)=\phi(g)\circ\phi(h)~\text{for all }g,h\in G.$$ 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 $S_n$, Lie groups $\mathrm{GL}_n(\mathbb{C})$, $\mathrm{SL}_n(\mathbb{C})$, $\mathrm{SO}_n(\mathbb{C})$, $\mathrm{Sp}_{2n}(\mathbb{C})$, matrix groups over finite fields $\mathrm{GL}_n(\mathbb{F}_q)$, $\mathrm{SL}_n(\mathbb{F}_q)$, $\mathrm{SO}_n(\mathbb{F}_q)$, $\mathrm{Sp}_{2n}(\mathbb{F}_q)$, etc.