Class function | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

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. The 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 . The 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 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: When K is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (9)

Related courses (2)

Related lectures (11)

Group Representations Theory Application Overview PHYS-314: Quantum physics II

Covers the application of group representations theory in quantum physics.

Group Repetition Theory: Character Representations PHYS-314: Quantum physics II

Explores character representations in group repetition theory, discussing irreducibility, equivalence, and associated values.

Tensor Product of Representations PHYS-314: Quantum physics II

Covers the tensor product of representations, symmetries of a triangle, irreducible representations, and practical applications in physics.

MATH-620(2): Topics in the theory of reductive algebraic groups, Lie algebras, and representation theory II

The topics addressed in this course are the structure theory of reductive algebraic groups, their associated Lie algebras, the related finite groups of Lie type, and the representation theory of all o

MATH-620(1): Topics in the theory of reductive algebraic groups, Lie algebras, and representation theory I

Class function

Representation theory of finite groups

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.

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).