Description of irreducible representations of $SL(2,mathbb R)$.

Let $rho: SL(2,mathbb R) to GL(V)$ be an irreducible representation of $SL(2,mathbb R)$. A paper I am reading says $rho$ can be described as a symmetric tensor product in $Sym^n(mathbb R^2)$, which has $A^n, A^{n-1}B,dots, B^n$ as a basis (over $mathbb R$), where $A=(1,0)^T$ and $B=(0,1)^T$.

But I am very confused about how this is actually connected to a representation $rho: SL(2,mathbb R) to GL(V)$. Say if I have a homogeneous polynomial

$$p(A,B):=c_0 A^n + c_1 A^{n-1}B + cdots c_n B^n,$$

then how can we view $p(A,B)$ as a representation
$rho: SL(2,mathbb R) to GL(V)$ literally?

It would be great if someone could also show me some reference to the proof of this result