Definitions and examples

Let G be a finite group. Then by a representation L of G one means a homomorphism x → Lx of G into the group of all non-singular linear transformations of some vector space H(L) onto itself. In the theory developed by Frobenius H(L) is a finite dimensional vector space over the complex numbers. The character χL of L is the complex valued function x → Trace(Lx). Two representations L and M are said to be equivalent, L ≃ M, if there exists a bijective linear transformation V from H(L) to H(M) such that V−1 MxV = Lx for all x in G. One shows that L and M are equivalent if and only if χL(x) ≡ χM (x). One defines the direct sum L ⊕ M of two representations in an obvious way as the direct sum of these vector spaces and shows that χL⊕U = χL + χM. A fundamental theorem asserts that every representation is a direct sum of representations L which are irreducible in the sense that H(L) has no proper subspaces which are carried into themselves by all Lx. Thus every character is a sum of characters of irreducible representations, i.e., of ‘irreducible characters’. The irreducible characters are orthogonal with respect to summation over the group and hence, in particular, are linearly independent. Thus every character is uniquely a finite linear combination of irreducible characters with non negative integer coefficients. It follows that to know all characters (or equivalently all equivalence classes of representations) it suffices to know the irreducible characters.