0
$\begingroup$

I have just started learning about group representations, and I've come across some confusing language:

$\rho : G \rightarrow GL(V)$ factors via a representation of $G \big/ ker(\rho)$

I can see how this representation "induces" a representation on the quotient, is that what it means? The word factor makes me think you can write a product or decompose something (similar to Maschke's theorem).

Just for context $G = BD_{4m}$ where $m$ is even and $\rho$ maps into $GL(2, \mathbb{C})$ with kernel $\{\pm 1 \}$, so $G \big/ ker(\rho) \cong D_{2m}$

  • 1
    By factors, they mean that the map $\rho$ can be written as a composition of the projection from $G$ to $G/\operatorname{ker}(\rho)$ and a representation $\rho'$ of that quotient. So in this way, $\rho$ is a product of maps, i.e. it has been factored. – 2017-01-25
  • 0
    I see, so does it have anything to do with irreducibility of the representation? – 2017-01-25
  • 1
    No, this is completely unrelated to the reducibility of the representation. – 2017-01-25
  • 0
    Thank you, you could put this as an answer and I'll accept it. – 2017-01-25

1 Answers 1

0

This is just the statement of the first isomorphism theorem. Since $$\rho: G\to GL(V)$$ is a homomorphism, there exists a unique (injective) homomorphism $G/\ker\rho\to GL(V)$.