shows how the characters of two given representations give rise to characters of related representations. By construction, there is a direct sum decomposition of . On characters, this corresponds to the fact that the sum of the last two expressions in the table is , the character of . Inner product and characters In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups: Definition (Class functions). A function is called a class function if it is constant on conjugacy classes of , i.e. Note that every character is a class function, as the trace of a matrix is preserved under conjugation. The set of all class functions is a –algebra and is denoted by . Its dimension is equal to the number of conjugacy classes of Proofs of the following results of this chapter may be found in [1], [2] and [3]. An inner product can be defined on the set of all class functions on a finite group: Orthonormal property. If are the distinct irreducible characters of , they form an orthonormal basis for the vector space of all class functions with respect to the inner product defined above, i.e. Every class function may be expressed as a unique linear combination of the irreducible characters . One might verify that the irreducible characters generate by showing that there exists no nonzero class function which is orthogonal to all the irreducible characters. For a representation and a class function, denote Then for irreducible, we have from Schur's lemma. Suppose is a class function which is orthogonal to all the characters. Then by the above we have whenever is irreducible. But then it follows that for all , by decomposability. Take to be the regular representation. Applying to some particular basis element , we get . Since this is true for all , we have It follows from the orthonormal property that the number of non-isomorphic irreducible representations of a group is equal to the number of conjugacy classes of Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that where non-negative integers, then is the character of the direct sum of the representations corresponding to Conversely, it is always