Karakterteorem

Karakterteorem is a fundamental concept in the field of representation theory, a branch of mathematics that studies the ways in which abstract algebraic structures can be represented as linear transformations of vector spaces. The term "karakterteorem" translates to "character theorem" in English, and it refers to a specific result that relates the characters of a representation to the structure of the group.

The character of a representation is a function that assigns to each element of the group a complex number, which is the trace of the corresponding matrix representation. The karakterteorem states that the characters of irreducible representations of a finite group form an orthonormal basis for the space of class functions on the group. This means that any class function can be uniquely expressed as a linear combination of the characters of irreducible representations.

The karakterteorem is a powerful tool in representation theory, as it allows one to decompose any representation into a direct sum of irreducible representations. This is known as the character decomposition or character expansion. The theorem is also closely related to the concept of the character table, which is a matrix that lists the characters of all irreducible representations of a group.

The karakterteorem was first proven by Georg Frobenius in 1896, and it has since become a cornerstone of representation theory. It has numerous applications in mathematics and physics, including in the study of Lie groups, quantum mechanics, and number theory. The theorem is also closely related to the concept of the character table, which is a matrix that lists the characters of all irreducible representations of a group.