Orthogonal projection formula

Statement

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$. Suppose $\phi :G\to GL(V)$ is a finite-dimensional linear representation of $G$ over $k$. Suppose further that $k$ is a sufficiently large field for $G$, viz $k$ contains all the $m^{th}$ roots of unity where $m$ is the exponent of $G$.

By Maschke's lemma, $\phi$ must be completely reducible i.e. it is the direct sum of irreducible representations. Suppose the irreducible representations are $\phi _{1},\phi _{2},\ldots ,\phi _{s}$ and their multiplicities are $a_{1},a_{2},\ldots ,a_{s}$ respectively. Then if $\chi _{i}$ is the character of $\phi _{i}$ and $\chi$ of $\phi$ we have:

$<\chi ,\chi _{i}>=a_{i}$

where

$<f_{1},f_{2}>={\frac {1}{|G|}}\sum _{g\in G}f_{1}(g)f_{2}(g^{-1})$

Also $\chi$ is orthogonal to the character of any irreducible linear representation not among the $\phi _{i}$s.

Note that when $k$ is not sufficiently large, it is no longer true that $<\chi ,\chi _{i}>$ is the multiplicity of $\phi _{i}$ in $\phi$. Actually we get:

$<\chi ,\chi _{i}>=a_{i}d$

where $d$ is a positive integer, which is actually the number of irreducible constituents of $\phi _{i}$ in a sufficiently large field containing $k$.

It still remains true, though, that the inner product is nonzero if and only if $\phi _{i}$ is a part of $\phi$.

Results used

• Maschke's averaging lemma
• First orthogonality theorem

Proof

The orthogonal projection formula follows from the first orthogonality theorem, which states that the characters of irreducible linear representations form an orthonormal set. The idea is that if:

$\phi =\sum _{i}a_{i}\phi _{i}$

Then by linearity of trace, we get:

$\chi =\sum _{i}a_{i}\chi _{i}$

Taking the inner product with any particular $\chi _{i}$, all the terms other than that $\chi _{i}$ give zero, and $\chi _{i}$ gives $a_{i}$ (this is where we use the first orthogonality theorem).

Further, if we take the character of any representation other than the $\phi _{i}$s, its inner product with all the $\chi _{i}$s is zero, and hence its inner product with $\chi$ is zero.

Consequences

Uniqueness of decomposition as a sum of irreducible representations

The orthogonal projection formula tells us that given a representation, we can determine the multiplicities of irreducible representations in it. Thus, a representation cannot be expressed as a sum of irreducible representations in more than one way.

Character determines the representation

A representation is determined upto equivalence, by its character. This is essentially because the character determines the multiplicities of the irreducible constituents, which in turn determines the representation uniquely. We can rephrase this as: any field of characteristic not dividing the order of a finite group, is a character-determining field for the group. That is, every representation over the field is uniquely determined by its character.

Further information: Non-modular implies character-determining

Regular representation as a sum of irreducible representations

The orthogonal projection formula can be used to show that the regular representation is:

$\sum d_{i}\chi _{i}$

where $\chi _{i}$ are the characters of irreducible linear representations, and $d_{i}$ is the degree of $\chi _{i}$.