[[Group representation theory MOC]]
# Group character

A **character** $\chi$ of a group $G$ over a [[field]] $\mathbb{K}$ is a map  $\chi : G \to \mathbb{K}$ that can be defined as the [[Trace]] of a [[Group representation]] $\mathfrak{X} : G \to \Vect_{\mathbb{K}}$. #m/def/rep 
$$
\begin{align*}
\chi(g) = \Tr \mathfrak{X}(g) = \sum_{j = 1}^{\dim \Gamma} \mathfrak{X}_{jj}(g)
\end{align*}
$$
Characters neatly summarize representations. See [[Character table]].

## Complex character
Since [[Trace]] is invariant under unitary equivalences, [[Unitary equivalence of representations|unitarily equivalent]] representations have the same character. 
If $\Gamma^\mu$ is an [[irrep]] then $\chi^\mu$ is an **irreducible character**.
The irreducible characters $\{ \chi^\mu \}$ are [[Group class function|class functions]] and [[Orthonormality of irreducible characters|form an orthonormal basis]] of [[Centre of the group ring|all such class functions within the group ring]] $Z(\mathbb{C}[G])$.



## Linear character
In the special case of a **linear character** the vector space is one-dimensional and thus the character is a homomorphism into the multiplicative group of $\mathbb{K}$,
i.e. a [[1-dimensional irrep|1-dimensional representation]].

## Properties

- [[Orthonormality of irreducible characters]]
- [[Character irreducibility criterion]]
- [[Irreducible character as function of an idempotent]]
- [[Finite group character values]]
- [[Tensor powers of a faithful representation contain all irreps]]

## See also

- [[Central character]]

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