The Schur-Weyl duality says that $n$th tensor power of a $d$-dimensional vector space over $\mathbb C$ is isomorphic to the direct sum of the tensor product of the the Weyl modules $V_\lambda$ and the Specht module $S_\lambda$, which are respectively the irreps of $\mathrm{GL}_d(\mathbb C)$ and that of the symmetric group $\mathfrak S_n$ on $n$ elements. Symbolically,
$$(\mathbb C^d)^{\otimes n}\cong \bigoplus_{\lambda\vdash n,\ell(\lambda)\leq d} V_\lambda\otimes S_\lambda.$$
as a $\mathrm{GL}_d(\mathbb C)\times\mathfrak{S}_n$-module.
This implies that as an $\mathfrak{S}_n$-module, $(\mathbb C^d)^{\otimes n}\cong\bigoplus_{\lambda\vdash n,\ell(\lambda)\leq d} \dim(V_\lambda) S_\lambda$ (i.e., each $S_\lambda$ has multiplicity $\dim(V_\lambda)$).
I wonder if there is a decomposition of $(\mathbb C^d)^{\otimes n}$ as an $\mathfrak{S}_n$-module into the permutation modules $M_\lambda$, and if there is one, what is the multiplicity of each $M_\lambda$? Here as in Sagan, Definition 2.1.5, a permutation of shape $\lambda$ is defined as the modules spanned by Young tabloids of shape $\lambda$.
I think it should be possible:
Consider, for example, a standard basis vector $v=e_{i_1}\otimes\ldots\otimes e_{i_n}$ for $i_1,\ldots,i_n\in\{1,\ldots,d\}$. We can define a Young tableau where each row contains indices $\ell_1,\ell_2,\ldots$ such that $i_{\ell_1}=i_{\ell_2}=\ldots$, and the group action maps one tableau to another of the same shape. However, I am not sure how to further justify my conjecture that such a decomposition exists and calculate the multiplicities. I have searched online, but most discussions I found were about decomposition permutation modules into Specht modules.
Best Answer
This seems to work fairly straightforwardly. Your map to Young Tabloids looks well defined and equivariant (we should additionally specify the way rows of the same width should be ordered).
Indeed it's not hard to find a copy of $M_\lambda$, where $\lambda=(\lambda_1,\dots,\lambda_h)$, is given by the span of the $e_{i_1}\otimes\dotsm\otimes e_{i_n}$ where $i_1,\dots,i_n$ comprises $\lambda_1$ lots of $x_1$, say, and $\lambda_2$ lots of $x_2$, etc. Counting the different possible choices of the $x_i$ which give the same tabloid I get a multiplicity of $$ m_\lambda = \frac{d!}{(d-h)!a_1!\dotsm a_k!}$$ Here $h$ is number of rows in the Young Tabloid (i.e. the length of $\lambda$) and $a_1$ is the number of rows with the longest length, $a_2$ is the number of rows with the second longest length and so on (so $a_1+ \dotsm + a_k =h$ for example).
Then we have $$(\mathbb{C}^d)^{\otimes n} = \bigoplus_{\lambda\vdash n,\ell(\lambda)\leq d}m_\lambda M_\lambda.$$
Note that there is at least one copy of $S_\lambda$ in each $M_\lambda$ but other than that I think this won't play super nicely with the Schur-Weyl decomposition. So the action of $GL_d(\mathbb{C})$ won't preserve this for example.