Character theory studies representations through the traces of the representing linear transformations.
The character of a finite-dimensional representation is the class function . Isomorphic representations have the same character.
An irreducible character is the character of an irreducible representation. Over the complex numbers, every finite-group character decomposes uniquely as a nonnegative integer combination of irreducible characters.
For a subgroup , the restriction of a -character to is the -character obtained by evaluating the same function only on elements of .
If is an irreducible character of a finite group and , thenEquality holds exactly when vanishes on .
Let have index two, and let be the nontrivial linear character of inflated to . For an irreducible character of , either is irreducible and , or the restriction is a sum of two distinct irreducible characters and .
A virtual character is an integer linear combination of characters, equivalently an element of the Grothendieck group of finite-dimensional representations.
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Character theory is a branch of mathematics, specifically within the field of representation theory of finite groups and algebra. It studies the characters of group representations, which are complex-valued functions that provide insight into the structure of the group. In essence, a character of a group representation is a function that assigns to each group element a complex number, which is the trace of the corresponding linear transformation in a representation.