6
$\begingroup$

Let $V$ be a nonzero finite dimensional representation, i.e we have a homomorphism $\rho\colon A\rightarrow \text{End}_k(V)$, of an algebra $A$. I have to show that there is an irreducible sub representation. This is how wanted to do that:

Let $v\in V$ and look at $W=\text{span}\{\rho(a)(v)\colon a\in A\}$. This is a lineair subspace of $V$ and by construction it is a sub representation. But it is not irreducible yet. I thought I should continue this process, so take again another vector in $W$ and consider the same construction of a sub representation. I don't understand how I should continue this or if this is going to help me solve this problem. I should also use somewhere the finiteness of the representation as it does not hold for infinite dimensional representations. I need help. Thanks.

$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

Hint: If $V$ has a proper non-trivial subrepresentation $W$, then $0<\dim W<\dim V$.

$\endgroup$
5
  • 1
    $\begingroup$ Ah I see, so if $W$ has a proper sub representation $W_2$ then $0<\text{dim}(W_2)<\text{dim}(W)$. If it has no proper sub representation, then we are already done. This process we can continue and it will terminate after finite steps, because the dimension of $V$ is finite. Thanks! $\endgroup$
    – Badshah
    Sep 13, 2014 at 19:40
  • $\begingroup$ Correct! Very well done, @Badshah! $\endgroup$
    – Jyrki Lahtonen
    Sep 13, 2014 at 19:40
  • $\begingroup$ @JyrkiLahtonen I just start reading rep theory and I was wondering, what's wrong with the argument that you take the intersection of all the sub-representation of $A$ in V? I believed intersection of sub-representation is also another sub-representation and the intersection is the smallest sub-representation so it has to be irreducible. I know there is something wrong with my argument because I don't use the finite dimension of $V$ but I can't find it. $\endgroup$
    – Khoa ta
    Jun 7, 2020 at 22:57
  • $\begingroup$ @Khoata That intersection is often just the trivial subrepresentation $\{0\}$, and we want a non-trivial subrepresentation. For example, when $V$ is a direct sum of several irreducible subrepresentations, then the intersection is automatically trivial. $\endgroup$
    – Jyrki Lahtonen
    Jun 8, 2020 at 4:24
  • $\begingroup$ @JyrkiLahtonen oh ok, thank you for the reply, I forgot that by def, irreducible representation has to be nonzero. $\endgroup$
    – Khoa ta
    Jun 8, 2020 at 5:43

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .