The character of a linear representation of G over a fieldK is always a class function with values in K. The class functions form the center of the group ringK[G]. Here a class function f is identified with the element .
class, function, confused, with, class, function, theory, mathematics, especially, fields, group, theory, representation, theory, groups, class, function, function, group, that, constant, conjugacy, classes, other, words, invariant, under, conjugation, such, f. Not to be confused with a class function in set theory In mathematics especially in the fields of group theory and representation theory of groups a class function is a function on a group G that is constant on the conjugacy classes of G In other words it is invariant under the conjugation map on G Such functions play a basic role in representation theory Contents 1 Characters 2 Inner products 3 See also 4 ReferencesCharacters editThe character of a linear representation of G over a field K is always a class function with values in K The class functions form the center of the group ring K G Here a class function f is identified with the element g G f g g displaystyle sum g in G f g g nbsp Inner products editThe set of class functions of a group G with values in a field K form a K vector space If G is finite and the characteristic of the field does not divide the order of G then there is an inner product defined on this space defined by ϕ ps 1 G g G ϕ g ps g displaystyle langle phi psi rangle frac 1 G sum g in G phi g overline psi g nbsp where G denotes the order of G and bar is conjugation in the field K The set of irreducible characters of G forms an orthogonal basis and if K is a splitting field for G for instance if K is algebraically closed then the irreducible characters form an orthonormal basis In the case of a compact group and K C the field of complex numbers the notion of Haar measure allows one to replace the finite sum above with an integral ϕ ps G ϕ t ps t d t displaystyle langle phi psi rangle int G phi t overline psi t dt nbsp When K is the real numbers or the complex numbers the inner product is a non degenerate Hermitian bilinear form See also editBrauer s theorem on induced charactersReferences editJean Pierre Serre Linear representations of finite groups Graduate Texts in Mathematics 42 Springer Verlag Berlin 1977 Retrieved from https en wikipedia org w index php title Class function amp oldid 1154446691, wikipedia, wiki, book, books, library,