Degree of a linear representation: Difference between revisions

Latest revision as of 16:16, 22 June 2008

This article gives a basic definition in the following area: linear representation theory
View other basic definitions in linear representation theory |View terms related to linear representation theory |View facts related to linear representation theory

Definition

Symbol-free definition

The degree of a linear representation is defined as the dimension of the vector space to which the representation's map is defined.

It also equals the value of the character at the identity element.

Definition with symbols

Suppose $(V,\rho )$ is a linear representation of a group $G$ over a field $k$ , i.e. we have a homomorphism $\rho :G\to GL(V)$ , where $V$ is a vector space over $k$ . Then, the degree of the representation $(V,\rho )$ is defined as the dimension of $V$ as a $k$ -vector space.

For a finite group, the degrees of irreducible representations are important numbers. Further information: degrees of irreducible representations

@@ Line 2: / Line 2: @@
 ==Definition==
+===Symbol-free definition===
 The degree of a linear representation is defined as the dimension of the vector space to which the representation's map is defined.
-Italso equals the value of the [[character of a linear representation|character]] at the identity element.
+It also equals the value of the [[defining ingredient::character of a linear representation|character]] at the identity element.
+===Definition with symbols===
+Suppose <math>(V,\rho)</math> is a linear representation of a group <math>G</math> over a field <math>k</math>, i.e. we have a homomorphism <math>\rho:G \to GL(V)</math>, where <math>V</math> is a vector space over <math>k</math>. Then, the degree of the representation <math>(V,\rho)</math> is defined as the dimension of <math>V</math> as a <math>k</math>-vector space.
+For a finite group, the degrees of irreducible representations are important numbers. {{further|[[degrees of irreducible representations]]}}