Let $V = \mathbb C^n$ and let $\{v_1,\cdots,v_n\}$ be a basis over $\mathbb C$. For $\ell\geq 1$, the symmetric group $S_\ell$ acts on the $\ell$-th tensor power $V^{\otimes \ell}$ by permuting the tensor entries: $\sigma(w_1\otimes \cdots\otimes w_\ell) = w_{\sigma^{-1}(1)}\otimes \cdots \otimes w_{\sigma^{-1}(\ell)}$ for any choice of $w_i \in V$.
This symmetric group representation on $V^{\otimes \ell}$ extends to an homomorphism of algebras $\rho: \mathbb C S_\ell \rightarrow \mbox{End}(V^{\otimes \ell})$. Since $\mathbb C S_\ell$ is a semisimple unital associative algebra, then $A = \rho(\mathbb C S_\ell)\subseteq \mbox{End}(V^{\otimes \ell})$ is also a semisimple algebra. Moreover, $A$ has finitely many finite-dimensional simple modules up to isomorphism. In fact, if $W$ is a simple $A$-module, then $W$ is also a $\mathbb C S_\ell$-module by the action $\sigma \cdot w = \rho(\sigma)\cdot w$. In particular, $W$ must also be simple as an $\mathbb C S_\ell$-module, by surjectivity.
Hence, all the inequivalent finite-dimensional simple $A$-modules must be a subset of the simple finite-dimensional $\mathbb C S_\ell$ modules given by $V_\lambda, \lambda \vdash \ell$.
Under the hyphotesis $\ell \leq n$, my question is the following: up to isomorphism, are all the simple finite-dimensional $A$-modules exactly all the simple finite-dimensional $\mathbb C S_\ell$-modules?
I am left with the impression that this is true. However, I cant come up with a proof of this fact. So far, I am betting that, because $V^{\otimes \ell}$ is large enough under the hyphotesis $\ell \leq n$, one can find at least a copy of every $V_\lambda$ inside $V^{\otimes \ell}$ for $\lambda \vdash \ell$. Hence, as a consequence, those will be also simple $A$-modules, since $A$ naturally acts on $V^{\otimes \ell}$. Is this intuition true? If so, how to construct such modules? I try to come up with a correspondence between Standard Young diagrams and tensors of the form $v_{i_1}\otimes \cdots \otimes v_{i_\ell}$ such that the span of "good choices" would give me the desired copy of $V_\lambda$, but so far I havent been sucessful.
Any help is very much appreciated, thank you!