A nice proof that the irreducible characters form a basis for class functions

 

Warning: This is a very shoddily organized post and can be vastly improved. The method of proof is still nice however.

Let G be a finite group of size g and k an algebraically closed field such that g \neq 0 in it. Let V_1,\dots, V_n be the irreducible representations of G over k and \chi_1,\dots,\chi_n the corresponding characters.

A class function f: G \to k is a set function such that f is constant on conjugacy classes. That is, for all g,h \in G, f(h^{-1}gh) = f(g). It is easy to see that any character is a class function by cyclicity of the trace function.

Let us denote by H the vector space of class functions and W the vector space of functions spanned by the \chi_i within H. It is natural to wonder about the size of W relative to H. In fact, the two vector spaces are equal!

To prove this, let us consider the algebra k[G]. By our hypothesis and the usual argument of Maschke’s theorem, this is a semisimple ring. That is, k[G] is semisimple as a G representation. Let r_G be the character attached to this so called regular representation. A simple computation shows us that r_1 = g and r_G = 0 otherwise.

Moreover, for any irreducible representation V_k, we see from last time that:

Hom_G(V_k,k[G]) = \frac{1}{g}\sum_g r_G(g^{-1})\chi_k(g) = \chi_k(1) = \dim V_k.

Therefore, by semisimplicity

k[G] \cong \oplus_k V_k^{\dim V_k}

as representations. Moreover, the orthogonality relations from last time also show that the \chi_k are linearly independent. That is, \dim W = \# irreducible characters.

Taking the endomorphism ring on both sides (as modules over k[G]), we obtain:

k[G] \cong M_{\dim V_k}(k)

as algebras. Considering dimensions, we have incidentally shown that g = \sum_k \dim V_k^2. Moreover, let us consider the central elements in k[G]. They are easily seen to exactly be the elements of the form \sum_g f(g)g for $f:G \to k$ a class function.

On the other hand, the central elements in M_n(k) are always just k. Therefore, the dimension of the center of k[G] (= vector space of class functions) is equal to the number of irreducible representations. Therefore, \dim W = \dim H and we are done.

The above map can be thought of as the representation of a class function in the basis defined by the functions c_k\overline{\chi_k} for some appropriate constants c_k. This is because the element:

\lambda_i = \sum_{g}\overline{\chi_i}g \in k[G]

maps to a a diagonal element in M_{\dim V_j}(k) that can be calculated by taking the trace. That is to say, the image of \lambda_i in $latex M_{\dim V_j}(k) is given by:

\frac{1}{\dim V_k}\sum_{g}\overline{\chi_i(g)}\chi_j(g) = \frac{g}{\dim V_k}\delta_{ij}

by the orthogonality relations. We see that c_k = \frac{\dim V_k}{g}.

 

 

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