Trivial linear representation: Difference between revisions

Revision as of 00:06, 3 November 2023

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

Let $G$ be a group, and $K$ be a field.

Then the map $ρ : G \to G L_{1} (K)$ sending each element of $G$ to the $1 \times 1$ identity matrix is a linear representation. It is called the trivial representation.

This representation is often denoted $1$

Character

The character of this representation is $1$ on all elements of the group.

@@ Line 6: / Line 6: @@
 Then the map <math>\rho: G \to GL_1(K)</math> sending each element of <math>G</math> to the <math>1 \times 1</math> identity matrix is a [[linear representation]]. It is called the '''trivial representation'''.
+This representation is often denoted <math>\mathbf{1}</math>
 ==Character==
 The [[character of a linear representation|character]] of this representation is <math>1</math> on all elements of the group.