Топтық теориядағы мінездемелер теориясы

Concept in mathematical group theory
the use of the term character theory in mathematics
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

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.

Let V be a finite dimensional vector space over a field F and let ρ : G → GL(V) be a representation of a group G on V. The character of ρ is the function χρ : G → F given by

where Tr is the trace. A character χρ is called irreducible or simple if ρ is an irreducible representation. The degree of the character χ is the dimension of ρ; in characteristic zero this is equal to the value χ(1). A character of degree 1 is called linear. When G is finite and F has characteristic zero, the kernel of the character χρ is the normal subgroup:

which is precisely the kernel of the representation ρ. However, the character is not a group homomorphism in general.

Characters are class functions, 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 'K' form a basis of the 'K' vector space of all class functions G → 'K'. Isomorphic representations have the same characters. Over a field of characteristic 0, two representations are isomorphic if and only if they have the same character. If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations. If a character of the finite group G is restricted to a subgroup H, then the result is also a character of H.
Every character value χ(g) is a sum of n m th roots of unity, 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 1=F = 'C', every such character value is an algebraic integer. If 1=F = 'C' and χ is irreducible, then is an algebraic integer for all x in G.
If F is algebraically closed and [[characteristic of a ring 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 ).

One may interpret the character of a representation as the "twisted" dimension of a vector space. 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 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. The (restriction to of the) character can be computed more explicitly by the Weyl character formula.