= Solution
The <symmetric group> acts on <tabloids> by permuting their entries; extending linearly gives the <permutation module> $M^\mu$. Its dimension is
$$
\boxed{\dim M^\mu=\frac{n!}{\prod_i\mu_i!}.}
$$
A tableau of type $\mu$ contains exactly $\mu_i$ copies of the label $i$. Fix a numbering of the $\lambda$-diagram's cells by $1,\ldots,n$. For a filling $T$, let the $i$th row set of an associated $\mu$-tabloid be the set of cell numbers whose labels in $T$ equal $i$. This is a bijection from type-$\mu$ fillings to $\mu$-tabloids.
Define $\sigma T$ by moving labels along the numbered positions: its label at cell $j$ is the old label at cell $\sigma^{-1}j$. The associated row set is then the image under $\sigma$ of the old row set. Thus the bijection is equivariant, and its linear extension makes $F[\mathcal T_{\lambda,\mu}]$ and $M^\mu$ <isomorphic representations>. Extend from group elements to the <group algebra> by linearity.
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