Grand orthogonality theorem

Statement over complex numbers
Suppose $$G$$ is a finite group. Let $$\mathbb{C}$$ denote the field of complex numbers. For each equivalence class of irreducible linear representation of $$G$$ over $$\mathbb{C}$$, choose a basis such that the representation is unitary, i.e., the image lies inside $$U(n,\mathbb{C})$$. Note that this can be done, because linear representation of finite group over complex numbers has invariant Hermitian inner product.

Now, consider the functions from $$G$$ to $$\mathbb{C}$$ obtained as the matrix entries for these representations. (There are $$n^2$$ functions for each representation of degree $$n$$).

Consider the usual Hermitian inner product on the space of complex-valued functions on $$G$$:

$$\langle f_1, f_2 \rangle = \frac{1}{|G|} \sum_{g \in G} f_1(g) \overline{f_2(g)}$$

Then:


 * 1) Any two matrix entry functions are orthogonal with respect to the inner product described above.
 * 2) The inner product of any matrix entry function with itself equals $$\frac{1}{n}$$ where $$n$$ is the degree of the representation from which it is picked.

Statement over general fields
Suppose $$G$$ is a finite group. Let $$k$$ be a splitting field for $$G$$ such that the characteristic of $$k$$ does not divide the order of $$G$$. For every equivalence class of irreducible linear representation of $$G$$ over $$k$$, choose a basis. Now consider the functions from $$G$$ to $$k$$ obtained as the matrix entries of these representations.

Consider the bilinear form for functions on the group:

$$\langle f_1, f_2 \rangle_G = \frac{1}{|G|} \sum_{g \in G} f_1(g)f_2(g^{-1})$$

Then:


 * 1) Matrix entries for distinct irreducible representations are orthogonal to each other, i.e., the inner product is zero.
 * 2) Matrix entries for the same irreducible representation have inner product $$1/n$$ iff the entries are transposes of each other, and have inner product zero otherwise.

Related facts

 * Sum of squares of degrees of irreducible representations equals order of group
 * Matrix entries form a basis for the space of all functions

Related facts

 * Character orthogonality theorem
 * Column orthogonality theorem

Facts used

 * 1) uses::Schur's lemma

Proof
We first provide the proof using the bilinear form for general splitting fields, then discuss how this can be used to deduce the proof for the Hermitian inner product if we use unitary matrices over the complex numbers.

Proof of orthogonality of matrix entries from inequivalent irreducible representations
This part of the proof does not require the field to be a splitting field.

Given: A finite group $$G$$, a field $$k$$ whose characteristic does not divide the order of $$G$$. For functions $$f_1,f_2: G \to k$$ define:

$$\langle f_1, f_2 \rangle_G = \frac{1}{|G|} f_1(g)f_2(g^{-1})$$

$$\varphi_1, \varphi_2$$ are inequivalent irreducible representations of $$G$$ over $$k$$ of degrees $$m,n$$ respectively.

Pick $$a,i \in \{ 1,2,\dots, m \}$$ and $$b,j \in \{ 1,2,\dots, n \}$$ ($$a,i$$ are allowed to be equal to each other, $$b,j$$ are allowed to be equal to each other).

Define:

$$f_1(g) := \varphi_1(g)_{ai}, \qquad f_2(g) := \varphi_2(g)_{jb}$$

To prove: $$\langle f_1, f_2 \rangle_G = 0$$

Proof:

Proof of orthogonality and norm of distinct matrix entries from the same representation when the field is algebraically closed
We assume the field to be algebraically closed. After that, we will discuss why the result holds for other splitting fields.

Given: A finite group $$G$$, an algebraically closed field $$k$$ whose characteristic does not divide the order of $$G$$. For functions $$f_1,f_2: G \to k$$ define:

$$\langle f_1, f_2 \rangle_G = \frac{1}{|G|} f_1(g)f_2(g^{-1})$$

$$\varphi$$ is an irreducible representation of degree $$n$$ and $$a,i,j,b$$ are all elements of $$\{ 1,2,\dots, n\}$$ (with the restriction that ).

Define:

$$f_1(g) := \varphi(g)_{ai}, \qquad f_2(g) := \varphi(g)_{jb}$$

To prove: If $$a = b$$ and $$i = j$$ (so $$f_1$$ and $$f_2$$ correspond to mutually transpose matrix entries), we have:

$$\langle f_1, f_2 \rangle_G = \frac{1}{n}$$

Otherwise, we have:

$$\langle f_1, f_2 \rangle_G = 0$$

Proof:

Proof for a splitting field
To prove the result for a splitting field, we embed the splitting field in an algebraically closed field, show the result there, and then note that all the inner product values remain the same upon restriction to a subfield containing the matrix entries.

Proof for unitary matrices
A unitary matrix is a matrix whose inverse is its conjugate-transpose. In particular, if $$\varphi(g)$$ is unitary, then:

$$\varphi(g)^{-1} = \overline{\varphi(g)^T}$$

In particular:

$$\varphi(g)^{-1}_{jb} = \overline{\varphi(g)}_{bj}$$

Thus:

$$\varphi(g)_{ai}\varphi(g^{-1})_{jb} = \varphi(g)_{ai}\overline{\varphi(g)}_{bj}$$

This equality allows us to go back and forth between the two formulations.