First applications of Schur’s lemma.Math 126 lecture 5.Recalling Schur’s lemma.The representation of G on Hom(V,W).Combining Schur’s lemma with averaging.Recall that Schur’s lemma says that if r and s are irreducible representations of G then(1) and (2) ifAverages of matrix elements.Averaging matrix elements of unitary representations.We have proved that in terms of a choice of bases we have:(5)To repeat: Since G is finite, the space of all complex valued functions on G is a finite dimensional vector space over the complex numbers. We call this space F(G) and put a scalar product on this space by declaring the scalar product of two functions be given byIf we are give a representation r of G on a vector space V and choose a basis of then for each position (ij) of the matrix describing r, we get a function on G. Suppose we have two inequivalent irreducible unitary representations r and s of G. Choose any matrix position for r and any matrix position for s. Then the functions we get are orthogonal relative to the scalar product. In symbols: whileThe character of a representation.The character of a direct sum is the sum of the characters.The characters of two inequivalent irreducible representations are orthogonal.Suppose that r and s are inequivalent irreducible representations of G.The character of an irreducible representation has length one.The character determines the representation.A character which has length one must be irreducible.dim Hom (V,W) when V is irreducible.Gdim Hom (U,V) in general, 1.Gdim Hom (U,V) in general,