Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures. χ 1 χ 2 . {displaystyle chi _{1}chi _{2}.} 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. Please note, that except for a few marked exceptions only finite groups will be considered in this article. We will also restrain 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 C . {displaystyle mathbb {C} .} 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 to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms. Let V {displaystyle V} be a K {displaystyle K} –vector space and G {displaystyle G} a finite group. A linear representation of a finite group G {displaystyle G} is a group homomorphism ρ : G → GL ( V ) = Aut ( V ) . {displaystyle ho :G o { ext{GL}}(V)={ ext{Aut}}(V).} That means, a linear representation is a map ρ : G → GL ( V ) {displaystyle ho :G o { ext{GL}}(V)} which satisfies ρ ( s t ) = ρ ( s ) ρ ( t ) {displaystyle ho (st)= ho (s) ho (t)} for all s , t ∈ G . {displaystyle s,tin G.} The vector space V {displaystyle V} is called representation space of G . {displaystyle G.} Often the term representation of G {displaystyle G} is also used for the representation space V . {displaystyle V.} The representation of a group in a module instead of a vector space is also called a linear representation. We write ( ρ , V ρ ) {displaystyle ( ho ,V_{ ho })} for the representation ρ : G → GL ( V ρ ) {displaystyle ho :G o { ext{GL}}(V_{ ho })} of G . {displaystyle G.} Sometimes we only use ( ρ , V ) , {displaystyle ( ho ,V),} if it is clear to which representation the space V {displaystyle V} belongs. In this article we will restrain ourselves to the study of finite-dimensional representation spaces, except for the last chapter. As in most cases only a finite number of vectors in V {displaystyle V} is of interest, it is sufficient to study the subrepresentation generated by these vectors. The representation space of this subrepresentation is then finite-dimensional. The degree of a representation is the dimension of its representation space V . {displaystyle V.} The notation dim ⁡ ( ρ ) {displaystyle dim( ho )} is sometimes used to denote the degree of a representation ρ . {displaystyle ho .}