Character orthogonality theorem

This fact is related to: linear representation theory
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This result is known as the first orthogonality theorem, character orthogonality theorem or row orthogonality theorem.

Statement

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$. Let ${\rho }_1$ and ${\rho }_2$ be two inequivalent irreducible linear representations of $G$ over $k$ and let ${\chi }_1$ and ${\chi }_2$ denote their characters. Then, the following are true:

${\sum }_{g\in G}{\chi }_1(g){\chi }_2(g^{-1})=0$

And:

${\sum }_{g\in G}{\chi }_1(g){\chi }_1(g^{-1})=d\left|G\right|$

where $d=1$ if the field $k$ is a sufficiently large field for $G$ (viz contains all the $m^{th}$ roots of $1$ where $m$ is the exponent of $G$).

When $k$ is not sufficiently large, $d$ is the number of irreducible constituents of $\varphi$ when taken over a sufficiently large field containing $k$.

In terms of inner product of class functions

For functions $f_1,f_2:G\to k$, define the following inner product:

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

Then, the character orthogonality theorem states that the characters of irreducible linear representations form an orthogonal set of elements, and further, if we are working over a sufficiently large field, they form an orthonormal set.

Note that by Maschke's lemma, the irreducible linear representations are precisely the indecomposable linear representations when the characteristic of $k$ does not divide the order of $G$, so we can replace irreducible in the above statement with indecomposable.

Consequences

• Orthogonal projection formula
• Column orthogonality theorem