= Solution
Let $V=\mathbb C^n$ and $m\geq0$. Identify the <symmetric power> $\operatorname{Sym}^mV$ with the degree-$m$ part of the symmetric algebra on the basis vectors $e_1,\ldots,e_n$, using formal variables $x_i$ for these vectors. Its monomial basis is
$$
x^a=x_1^{a_1}\cdots x_n^{a_n},\qquad a_i\geq0,\qquad\sum_i a_i=m.
$$
The diagonal <maximal torus> of the <unitary group> acts on this monomial by the character $z_1^{a_1}\cdots z_n^{a_n}$. Distinct tuples give distinct characters, so each <weight space> is one-dimensional.
Suppose $M$ is a nonzero invariant complex subspace. For a nonzero vector $w\in M$, select a monomial with nonzero coefficient and extract it by torus averaging:
$$
\int_T z_1^{-a_1}\cdots z_n^{-a_n}\,t\cdot w\,dt.
$$
<Fourier orthogonality> makes this a nonzero scalar multiple of $x^a$. It belongs to $M$, so $M$ contains a monomial.
Differentiating the <unitary group> action makes $M$ invariant under $\mathfrak u(n)$ and hence under its complexification $\mathfrak{gl}_n(\mathbb C)$. The matrix unit $E_{ij}$ replaces an occurrence of $e_j$ by $e_i$, so its action on the symmetric algebra is
$$
E_{ij}\cdot x^a=x_i\partial_{x_j}x^a
=a_jx^{a+e_i-e_j}.
$$
Whenever $a_j>0$, this transfers one unit of degree from coordinate $j$ to coordinate $i$ with a nonzero coefficient. Starting with any monomial, such transfers can produce every other tuple of nonnegative integers summing to $m$. Thus $M$ contains every monomial and equals $\operatorname{Sym}^mV$.
Consequently
$$
\boxed{\operatorname{Sym}^m\mathbb C^n\text{ is irreducible for every }m\geq0.}
$$
For $m=0$ this is the one-dimensional trivial representation, and for $n=1$ every symmetric power is also one-dimensional. This <weight-transfer proof of irreducibility of symmetric powers> uses neither the <Weyl character formula> nor the <Weyl dimension formula>, so no unproved dimension formula enters the argument.
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