## Matrix Entry Functions (Pt. II)

**Point of post: **This post is a continuation of this one.

*Matrix Entry Functions*

Suppose that for each we had created some matrix representation . We can then produce elements of the group algebra . Indeed, for a general matrix let denote the entry of . Then, for each define by . We call these the *matrix entry functions induced by . *Then, considering that it makes sense to speak of the span of this set, in particular denote by . We now derive some fundamental facts about .

**Theorem: *** separates points. (By separates points we mean that for any distinct there exists some such that and ).*

**Proof: **Let . Define by where denotes the usual left representation of into and is the usual inner product on the group algebra. Now, from previous theorem we know that and thus . But, recall that is orthonormal with respect to this inner product and thus clearly and for . Thus, the theorem will be proven if we can show that . To do this we merely note that since is a representation of we know that

for some finite (and the were the one’s fixed at the outset of this post). In particular, there exists some fixed unitary map such that for each one has that

Thus, if denotes some ordering of the basis and denotes the canonical ordered basis for (where these are the representation spaces for the fixed ) then implies that for some fixed matrix one has that

but is such that this can be rewritten as

Note though that by definition and using the bilinearity of the inner product, and the fact that is orthonormal we see that

note though that upon comparison of with the entry of

we may conclude that from where the conclusion follows.

We now show that has the property that the pointwise product of two elements of is an element of .

**Theorem: ***Let , then where .*

**Proof: **Let be arbitrary and let be the prefixed representatives of respectively. Recall then is a representation. This then implies that

for some finite . In particular, if denotes the usual ordered tensor basis for and the usual ordered direct sum basis for we have that for some fixed matrix ‘

for every . But because of how were chosen we may rewrite this as

where the left side denotes the Kronecker product. The result then follows by comparing the entries of the left and right and sides of this equation.

**References: **

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.

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