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 a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz
Mar 26
new correction: Misspelled name by DavidSteinsaltz
Mar 21
new correction: underlinetypo by Filipe
Mar 19
new correction: cocycle pro cocyle by pahio
Mar 7
new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier
new image: expected waiting time by robert_dodier
new image: plot W(t) = P(waiting time <= t) by robert_dodier