## Matrix Entry Functions Form an (almost) Orthonormal Basis

**Point of post: **In this post we derive the result that the matrix entry functions form an orthonormal basis for the group algebra, thus deriving the fundamental result that .

*Motivation*

In our last post we showed how fixing, for each , some representative and some orthonormal ordered basis enabled us to form ‘matrix entry functions’ which were elements of the group algebra . We further derived some important properties about the span of these matrix entry functions (that it is closed under pointwise product and that it separates points). In this post we take this further and show, using a simple case of the Stone-Weierstrass theorem, that . Moreover, we show that in the usual inner product on we have that the set of matrix entry functions is almost (up to a scalar factor) orthonormal. Thus, it will follow that the set of matrix entry functions is an orthonormal basis for and thus we will derive the fundamental result that and much sharper that .

*Orthogonality of the Matrix Entry Functions*

Let be a finite group and be as before . We claim that in the usual inner product on the group algebra is (up to a scalar factor) orthonormal. More formally:

**Theorem: ***For any one has the relation*

**Proof: **Fix some . ** **We then let be any linear map. We define then

We note then that is an intertwinor for and . Indeed, for any

It then follows from Schur’s lemma that if (i.e. if ) and otherwise. If then we note then that for the one has that

and thus either way . Note that this was true for *any *linear mapping . In particular, fix and and let the entry of (thought of as a matrix) be given by . We then compute (by just brute force) that the entry of is

But, noting that we see that and thus the the entry of is . Thus, comparing these two forms for the entry of gives that and since were arbirary the conclusion follows.

**Corollary: *** is a linearly independent subset of .*

* *

**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

[…] Point of post: This is a continuation of this post. […]

Pingback by Representation Theory: Matrix Functions Form an (almost) Orthonormal Basis (Pt. II) « Abstract Nonsense | February 23, 2011 |

[…] our last series of posts we saw an interesting technique. We saw the interesting idea that if we want to prove the […]

Pingback by Representation Theory: Class Functions « Abstract Nonsense | February 24, 2011 |

[…] certain relations between the pairwise inner product (as elements of the group algebra) of matrix entry functions and irreudcible characters. We shall use these relations to compute the convolution of matrix […]

Pingback by Representation Theory: Using Orthogonality Relations to Compute Convolutions of Characters and Matrix Entry Functions « Abstract Nonsense | March 2, 2011 |

[…] associated with . But, we then have that for each . Thus, the last step of may, in light of the orthogonality of the matrix entry functions, […]

Pingback by Representation Theory: A Characterization of Real, Complex, and Quaternionic Irreps « Abstract Nonsense | March 23, 2011 |

[…] carries the orthonormal basis to an orthonormal basis, and thus is unitary as claimed. The conclusion follows from previous […]

Pingback by Representation Theory: Decomposing the Group Algebra Into the Direct Sum of Matrix Algebras « Abstract Nonsense | April 6, 2011 |

[…] algebras. We shall use this fact to derive an interesting fact about the group algebra. Namely, we know that for every choice of matrix entry functions one has that the group algebra is a direct sum of […]

Pingback by Representation Theory: Consequence of the Decomposition of the Group Algebra Into Matrix Algebras « Abstract Nonsense | April 9, 2011 |