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Concept

Representation theory of finite groups

Summary

The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over \Complex. Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra

Official source

https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups

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Restricting representations of classical algebraic groups to maximal subgroups

Mikaël Cavallin

Fix an algebraically closed field

K

having characteristic

p\geq 0

and let

Y

be a simple algebraic group of classical type over

K.

Also let

X

be maximal among closed connected subgroups of

Y

and consider a non-trivial

p

-restricted irreducible rational

KY

-module

V.

In this thesis, we investigate the triples

(Y,X,V)

such that

X

acts with exactly two composition factors on

V

and see how it generalizes a question initially investigated by Dynkin in the

1950

s and then studied by numerous mathematicians. In particular, we study the natural embeddings of

\mbox{SO}_{2n}(K)

in both

\mbox{Spin}_{2n+1}(K)

and

\mbox{SL}_{2n}(K)

and obtain results on the structure of certain Weyl modules.

EPFL2015

Representation theory of finite groups

Restricting representations of classical algebraic groups to maximal subgroups

Endo-trivial modules: a reduction to p'-central extensions

Restricting representations of classical algebraic groups to maximal subgroups

Endo-trivial modules: a reduction to p'-central extensions