Applications[edit]

Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow $2$ -subgroup.

Characters are , that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible characters of a given group $G$ into a field $F$ form a basis of the $F$ -vector space of all class functions $G \to F$ .

class functions

representations have the same characters. Over a field of characteristic $0$ , two representations are isomorphic if and only if they have the same character.^[1]

Isomorphic

If a representation is the of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations.

direct sum

If a character of the finite group $G$ is restricted to a $H$ , then the result is also a character of $H$ .

subgroup

Every character value $χ (g)$ is a sum of $n$ $m$ -th , where $n$ is the degree (that is, the dimension of the associated vector space) of the representation with character $χ$ and $m$ is the order of $g$ . In particular, when $F = C$ , every such character value is an algebraic integer.

roots of unity

If $F = C$ and $χ$ is irreducible, then
$[G:C_{G}(x)]{\frac {\chi (x)}{\chi (1)}}$ is an for all $x$ in $G$ .

algebraic integer

If $F$ is and $char (F)$ does not divide the order of $G$ , then the number of irreducible characters of $G$ is equal to the number of conjugacy classes of $G$ . Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of $G$ (and they even divide $[G : Z (G)]$ if $F = C$ ).

algebraically closed

Decomposing an unknown character as a linear combination of irreducible characters.

Constructing the complete character table when only some of the irreducible characters are known.

Finding the orders of the centralizers of representatives of the conjugacy classes of a group.

Finding the order of the group.

"Twisted" dimension[edit]

One may interpret the character of a representation as the "twisted" dimension of a vector space.^[3] Treating the character as a function of the elements of the group $χ (g)$ , its value at the identity is the dimension of the space, since $χ (1) = Tr(ρ (1)) = Tr(I V) = dim(V)$ . Accordingly, one can view the other values of the character as "twisted" dimensions.

One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the $j$ -invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.^[3]

Irreducible representation § Applications in theoretical physics and chemistry

a combinatorial generalization of group-character theory.

Association schemes

introduced by A. H. Clifford in 1937, yields information about the restriction of a complex irreducible character of a finite group $G$ to a normal subgroup $N$ .