# Show that the representation $\mathbb Z\ni a\mapsto\begin{pmatrix}1& a\\0&1\end{pmatrix}$ is not completely reducible

Let $$\rho : \mathbb Z \to \mathrm{GL}_2(\mathbb C)$$ be the representation defined by $$\rho(1) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$. I'd like to show that $$\rho$$ is not completely reducible.

1. I have one preliminary question (which is probably a silly one) - for what vector space $$V$$ is $$\mathrm{GL}(V) \cong \mathrm{GL}_2(\mathbb C)$$?

Firstly, I noted that $$\rho(1)$$ has an eigenvector, so the representation is not irreducible. So if it were completely reducible, it would have to break up as a direct sum of two $$1$$-dimensional sub representations. But a 1-dimensional subrep is given by an eigenvector - but $$\rho$$ only has one eigenvalue, which has a $$1$$-dimensional eigenspace. So this can't happen.

2. Is this reasoning OK?

Once I've shown that the representation isn't irreducible, the problem is equivalent to showing that $$\rho(1)$$ cannot be diagonalised (which I've done by showing that the sum of the dimensions of the eigenspaces is $$1$$, not 2).

Depending on the answer to question (1), I could have reduced (excuse the pun) the amount of work by considering Jordan Normal Form ($$\rho(1)$$ is in JNF but isn't diagonal, so isn't diagonalisable).

• I think I am being silly with question i). $V = \mathbb C^2$ as a $\mathbb C$-vector space, right?
– Matt
Mar 17, 2012 at 16:01
• So the Jordan Normal Form argument does apply
– Matt
Mar 17, 2012 at 16:02
• Indeed, the vector space is $\mathbb C^2$., and yes, the JNF argument does work, too. Mar 17, 2012 at 16:04
• @Matt : The only such $V$'s you will find are those who are isomorphic to $\mathbb C^2$. This is the classical example where we have a counter-example to justify representation theory working only with finite groups. When in infinite group, the whole idea of computing characters to work with the representation is pointless. Mar 17, 2012 at 16:04
• @Patrick: Well... pointless is a bit too strong :) For example, in the correct context, the finite dimensional representation theory of many infinite groups (like semisimple Lie groups, say) is completely controled by characters. Mar 17, 2012 at 16:15

As you already noted yourself in the comments you can take $V=\mathbb{C}^2$ and then a Jordan normal form argument is perfectly fine.