Character Theory - Properties

Properties

• 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, 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 mth 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 F is the field of complex numbers, every such character value is an algebraic integer.
• If F is the field of complex numbers, and χ is irreducible, then is an algebraic integer for each x in G.
• If F is algebraically closed and char(F) does not divide |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 the index of the center of G in G if F = C).