Character orthogonality theorem

This fact is related to: linear representation theory
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This article describes an orthogonality theorem. View a list of orthogonality theorems

Name

This result is known as the first orthogonality theorem, character orthogonality theorem or row orthogonality theorem.

Statement

Statement over complex numbers

Let $G$ be a finite group and $C$ denote the field of complex numbers. Let $\bar{z}$ denote the complex conjugate of $z$ . Then, if $ρ_{1}$ and $ρ_{2}$ are two inequivalent irreducible linear representations, and $χ_{1}$ and $χ_{2}$ are their characters, we have:

$\sum_{g \in G} χ_{1} (g) \bar{χ_{2} (g)} = 0$

and:

$\sum_{g \in G} χ_{1} (g) χ_{1} (g) = | G |$

Statement over complex numbers in terms of inner product of class functions

Consider the space of complex-valued functions $G \to C$ . This is a $C$ -vector space in a natural way, with basis being the indicator functions of elements of $G$ . Consider the Hermitian inner product on this vector space given by:

$⟨ f_{1}, f_{2} ⟩ = \frac{1}{| G |} \sum_{g \in G} f_{1} (g) \bar{f_{2} (g)}$

Then, the characters form an orthonormal set of functions with respect to this basis.

Statement over general fields

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$ . Let $ρ_{1}$ and $ρ_{2}$ be two inequivalent irreducible linear representations of $G$ over $k$ and let $χ_{1}$ and $χ_{2}$ denote their characters. Then, the following are true:

$\sum_{g \in G} χ_{1} (g) χ_{2} (g^{- 1}) = 0$

And:

$\sum_{g \in G} χ_{1} (g) χ_{1} (g^{- 1}) = d | G |$

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

When $k$ is not sufficiently large, $d$ is the number of irreducible constituents of $χ_{1}$ when taken over a splitting field containing $k$ .

Statement over general fields in terms of inner product of class functions

For functions $f_{1}, f_{2} : G \to k$ , define the following inner product:

$⟨ f_{1}, f_{2} ⟩ = \frac{1}{| G |} \sum_{g \in G} f_{1} (g) f_{2} (g^{- 1})$

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