Grand orthogonality theorem

This article gives the statement, and possibly proof, of a basic fact in linear representation theory.
View a complete list of basic facts in linear representation theory OR View all facts related to linear representation theory

This article describes an orthogonality theorem. View a list of orthogonality theorems

Statement

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 $U(n,\mathbb{C})$. 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 $G$:

$⟨f_1,f_2⟩=\frac{1}{\left|G\right|}{\sum }_{g\in G}{f}_1(g)\overline{f_2(g)}$

Then:

1. The total number of functions arising from matrix entries equals the order of the group $G$
2. Any two such functions are orthogonal with respect to the inner product described above.
3. 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:

$⟨f_1,f_2{⟩}_G=\frac{1}{\left|G\right|}{\sum }_{g\in G}{f}_1(g)f_2(g^{-1})$

Then:

1. The total number of functions arising from matrix entries equals the order of the group $G$
2. Any two such functions are orthogonal with respect to the bilinear form described above.
3. The inner product of any matrix entry function with itself equals $\frac{1}{n}$ where $d$ is the degree of the representation from which it is picked.

Related facts

• Character orthogonality theorem
• Column orthogonality theorem

Facts used

1. Schur's lemma
2. Trace of inverse is complex conjugate of trace

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:

$⟨f_1,f_2{⟩}_G=\frac{1}{\left|G\right|}{f}_1(g)f_2(g^{-1})$

${\phi }_1,{\phi }_2$ 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:

$f_1(g):={\phi }_1(g{)}_{ai},f_2(g):={\phi }_2(g{)}_{jb}$

To prove: $⟨f_1,f_2{⟩}_G=0$

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	Denote by $E_{ij}$ the $m\times n$ matrix with a 1 in the $i{j}^{th}$ entry and 0s elsewhere.	--	--	--	--
2	Define $F_{ij}=\frac{1}{\left|G\right|}{\sum }_{g\in G}{\phi }_1(g)E_{ij}{\phi }_2(g^{-1})$. $F_{ij}$ is a $m\times n$ matrix.		${\phi }_1$ has degree $m$, ${\phi }_2$ has degree $n$, so the matrix multiplication makes sense.	Step (1)	--
3	$F_{ij}$ is a homomorphism of representations from ${\phi }_2$ to ${\phi }_1$			Step (2)	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
4	$F_{ij}$ is the zero matrix.	Fact (1) (note that the roles of ${\phi }_1,{\phi }_2$ are interchanged from the statement of Fact (1))	${\phi }_1,{\phi }_2$ are inequivalent irreducible representations.	Step (3)
5	The $(ab{)}^{th}$ entry of $F_{ij}$ is zero.			Step (4)
6	The $(ab{)}^{th}$ entry of $F_{ij}$ equals $\frac{1}{\left|G\right|}{\sum }_{g\in G}{\phi }_1(g{)}_{ai}{\phi }_2(g^{-1}{)}_{jb}$.			Step (2)	Follows by simplifying the matrix multiplication.
7	$\frac{1}{\left|G\right|}{\sum }_{g\in G}{\phi }_1(g{)}_{ai}{\phi }_2(g^{-1}{)}_{jb}=0$, so $⟨f_1,f_2{⟩}_G=0$.		Definitions of $f_1,f_2$, inner product	Steps (5), (6)	Step-combination direct for first version, use definitions of $f_1,f_2$ for second version.

Proof of orthonormality 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:

$⟨f_1,f_2{⟩}_G=\frac{1}{\left|G\right|}{f}_1(g)f_2(g^{-1})$

$\phi$ 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:

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