Matrix Functions Form an (almost) Orthonormal Basis (Pt. II)
Point of post: This is a continuation of this post.
Matrix Entry Functions Span the Group Algebra
We now show that . To do this we prove a result similar to that of the Stone-Weierstrass theorem for finite sets. Namely:
Theorem: Let be some finite set and a subalgebra of which separates points, then . (here the algebra structure is the usual pointwise scalar multiplication, pointwise addition, and pointwise product of functions).
Proof: Note that is a basis for where . Note though, for each we are guaranteed some such that and . It’s evident though that
and so in particular since is a subalgebra we have that for every and since is, in particular, a subspace we have that . The conclusion follows.
From this we pretty easily derive the desired result. Namely:
Proof: We’ve proven in previous posts that is a subalgebra of (this equality is as a set, since the group algebra has a different multiplicative structure than pointwise multiplication) which separates points, and thus the conclusion follows by the previous theorem.
Corollary: is a basis for and thus in particular and
Note though that since we also have the trivial irrep that the above implies that, for example, any non-trivial irrep has the quality that . We shall use this to derive many, many more interesting result as the theory progesses.
1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print