You are here
Homecharacter
Primary tabs
character
Let $\rho:G\longrightarrow\operatorname{GL}(V)$ be a finite dimensional representation of a group $G$ (i.e., $V$ is a finite dimensional vector space over its scalar field $K$). The character of $\rho$ is the function $\chi_{V}:G\longrightarrow K$ defined by
$\chi_{V}(g):=\operatorname{Tr}(\rho(g))$ 
where $\operatorname{Tr}$ is the trace function.

$\chi_{V}(g)=\chi_{V}(h)$ if $g$ is conjugate to $h$ in $G$. (Equivalently, a character is a class function on $G$.)

If $G$ is finite, the characters of the irreducible representations of $G$ over the complex numbers form a basis of the vector space of all class functions on $G$ (with pointwise addition and scalar multiplication).

Over the complex numbers, the characters of the irreducible representations of $G$ are orthonormal under the inner product
$(\chi_{1},\chi_{2}):=\frac{1}{G}\sum_{{g\in G}}\overline{\chi_{1}(g)}\chi_{2% }(g)$
Mathematics Subject Classification
20C99 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis
Apr 20
new image: informationtheoreticdistributedmeasurementdds.png by rspuzio
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia