Definition and first properties
The character of a representation ρ is the function χ: G → C given by χ(g) = tr ρ(g), the trace of the matrix. Trace is basis-independent, so χ does not depend on the chosen basis, and isomorphic representations have equal characters. Two facts fall out immediately: χ(e) = tr(id) = dim V is the degree, and since tr(ABA⁻¹) = tr(B), we get χ(hgh⁻¹) = χ(g). A character is therefore a class function — constant on each conjugacy class.
More properties, all proven by linear algebra. For a finite group ρ(g) has finite order, so it is diagonalizable with roots of unity on the diagonal; hence χ(g⁻¹) = conjugate of χ(g), and |χ(g)| ≤ dim V. The character of a direct sum is the sum of characters, and the character of a tensor product is the product: χ_{V⊗W}(g) = χ_V(g)·χ_W(g). Characters turn the algebra of representations into ordinary arithmetic of functions.
Worked example: the character table of S_3
S_3 has three conjugacy classes: the identity {e}, the three transpositions, and the two 3-cycles. There are exactly three irreducibles (we will see in guide 4 why the count of irreducibles equals the count of classes). The trivial character is constantly 1; the sign character sends even permutations to 1 and odd to −1; and there is a 2-dimensional standard representation.
Conjugacy classes of S_3 and their sizes:
class: e (12) (123)
size: 1 3 2
Character table (rows = irreducibles, columns = classes):
e (12) (123)
trivial 1 1 1
sign 1 -1 1
standard 2 0 -1
Getting the standard row: S_3 permutes coordinates of C^3; the
permutation character is (3, 1, 0) [fixed points of each class].
Subtract the trivial (1,1,1): (3,1,0) - (1,1,1) = (2, 0, -1).
That 2-dim complement is the standard irreducible — its row, read off.
Degree check (sum of squares = |G|): 1^2 + 1^2 + 2^2 = 6 = |S_3|. OKNotice the column of fixed points trick: any action of G on a finite set gives a permutation representation, and its character at g is simply the number of points g fixes. Subtracting off the trivial part is how the standard representation appeared. This is the everyday way characters get computed in practice.