Character of a linear representation: Difference between revisions

Latest revision as of 22:00, 6 November 2023

This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space

This article gives a basic definition in the following area: linear representation theory
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Definition

Definition in terms of linear representation as a homomorphism

Let $G$ be a group and $ρ : G \to G L (V)$ be a finite-dimensional linear representation over a field $k$ . Then, the character of $ρ$ is the composite $T r \circ ρ$ where $T r$ is the trace map from $G L (V)$ to $k$ .

Definition in terms of linear representation as an algebra map

Elementary properties

Characters of representations are class functions, that is, they are constant on each conjugacy class of the group.

A character is called irreducible if its corresponding representation is an irreducible representation.

Character tables

The notion of characters leads to that of a character table of a group; given a group $G$ , the character table lists the value of each of the irreducible characters of the group on each conjugacy class (and thus each element, since the character is a class function)

Examples

The trivial representation of a group admits the trivial character - equal to $1$ on each element.

The character of the regular representation of a finite group $G$ takes the value $| G |$ at the identity element and zero elsewhere. This is because the character of the corresponding permutation matrix is the number of fixed points, and multiplication by a non-identity element has no fixed points.

The sign representation on the symmetric group $S_{n}$ has character $1$ on even elements, $- 1$ on odd elements.

The standard representation of the symmetric group $S_{n}$ has character $χ (g)$ equal to the number of fixed points of the permutation $g \in S_{n}$ minus one.

@@ Line 7: / Line 7: @@
 ===Definition in terms of linear representation as a homomorphism===
-Let <math>G</math> be a group and <math>\rho: G \to GL(V)</math> be a [[finite-dimensional linear representation]] over a field <math>k</math>. Then, the character of <math>\rho</math> is the composite <math>Tr \circ \rho</math> where <math>Tr</math> is the trace map from <math>GL(V)</math> to <math>k</math>.
+Let <math>G</math> be a [[group]] and <math>\rho: G \to GL(V)</math> be a [[finite-dimensional linear representation]] over a field <math>k</math>. Then, the character of <math>\rho</math> is the composite <math>Tr \circ \rho</math> where <math>Tr</math> is the trace map from <math>GL(V)</math> to <math>k</math>.
 ===Definition in terms of linear representation as an algebra map===
+==Elementary properties==
+Characters of representations are [[class function|class functions]], that is, they are constant on each [[conjugacy class]] of the group.
+A character is called [[irreducible character|irreducible]] if its corresponding representation is an [[irreducible representation]].
+==Character tables==
+The notion of characters leads to that of a [[character table]] of a group; given a group <math>G</math>, the character table lists the value of each of the [[irreducible character|irreducible characters]] of the group on each conjugacy class (and thus each element, since the character is a class function)
+==Examples==
+* The [[trivial linear representation|trivial representation]] of a group admits the trivial character - equal to <math>1</math> on each element.
+* The character of the [[regular representation]] of a [[finite group]] <math>G</math> takes the value <math>|G|</math> at the identity element and zero elsewhere. This is because the character of the corresponding [[permutation matrix]] is the number of fixed points, and multiplication by a non-identity element has no fixed points.
+* The [[sign representation on symmetric group|sign representation]] on the [[symmetric group]] <math>S_n</math> has character <math>1</math> on even elements, <math>-1</math> on odd elements.
+* The [[standard representation of the symmetric group|standard representation]] of the [[symmetric group]] <math>S_n</math> has character <math>\chi(g)</math> equal to the number of fixed points of the permutation <math>g \in S_n</math> minus one.