= Solution
Give $V_n$ the action $(g\cdot P)(x,y)=P((x,y)g)$, where $(x,y)$ is a row vector. Substitution preserves homogeneous degree, and composition agrees with <matrix multiplication>, so this is a <complex representation> of <SU(2) group>. The diagonal <torus> $\operatorname{diag}(z,z^{-1})$ acts on $x^{n-j}y^j$ with weight $z^{n-2j}$; these weights are distinct.
A nonzero <invariant subspace> is invariant under the <torus>, so finite-dimensional Fourier projection onto its <weight spaces> supplies a nonzero <monomial>. Differentiating the <group action> and complexifying supplies the operators $E=x\partial_y$ and $F=y\partial_x$. Their repeated application moves between every successive <monomial>, with nonzero coefficients until an endpoint is reached:
$$
E(x^{n-j}y^j)=j x^{n-j+1}y^{j-1},\qquad
F(x^{n-j}y^j)=(n-j)x^{n-j-1}y^{j+1}.
$$
The <invariant subspace> therefore contains the whole <monomial basis>. Thus
$$
\boxed{V_n\text{ is irreducible and }\dim V_n=n+1.}
$$
For $n=0$ this is the one-dimensional <trivial representation>. <Torus> weight projections are legitimate because an invariant complex subspace is closed in finite dimension.
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