Definition
Representation
A (linear) representation of a group $G$ (Definition~Group (Classical)) on a vector space $V$ over a field $k$ is a group homomorphism (Definition~Group Homomorphism)
\[
\rho : G \to \mathrm{GL}(V).
\]
We say $(V, \rho)$ is a $G$-representation, or simply a $G$-module. The dimension of the representation is $\dim V$.
Depends on
Used in
- Subrepresentation
- Irreducible Representation
- Maschke's Theorem
- Regular Representation
- Equivalent Categories kG-Mod & Rep G
- Subrepresentation and Invariant Subspace
- Irreducible Representation
- Morphism of Representations
- Direct Sum of Representations
- Complete Reducibility
- Maschke's Theorem for Compact Groups
- Character
- Proposition
- Inner Product
- Representation of a Lie Group
Dependency Graph
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