The skew multiplicity module $K^{\lambda \mathrm{/}\mu }$ over $\mathbb{C}$

Let $\mu \subseteq \lambda$ be partitions, put $m:=\mathrm{\mid }\mu \mathrm{\mid }$ and $r:=\mathrm{\mid }\lambda \mathrm{\mid }-m$, and let $S^{\mu }$ and $S^{\lambda }$ be the complex Specht modules. Use the block subgroup $H_{m,r}\le S_{m+r}$ and the group identification ${\iota }_{m,r}:S_m\times S_r\to H_{m,r}$ of The restriction coproduct on the graded symmetric-group character ring. Restrict $S^{\lambda }$ to $H_{m,r}$ and pull the action back along ${\iota }_{m,r}$; write the resulting finite-dimensional complex $S_m\times S_r$-module as $V_{m,r}^{\lambda }$ (The sign representation of $S_n$ and the restriction ${\mathrm{Res}}_H^G(V)$ of a representation to a subgroup, A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree).

View $S^{\mu }$ and $V_{m,r}^{\lambda }$ as left $\mathbb{C}[S_m]$-modules via the correspondence between group actions and group-ring modules (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures). The skew multiplicity module is

$K^{\lambda \mathrm{/}\mu }:={\mathrm{Hom}}_{\mathbb{C}[S_m]}(S^{\mu },V_{m,r}^{\lambda }),$

the space of $\mathbb{C}[S_m]$-module homomorphisms (The abelian group ${\mathrm{Hom}}_R(M,N)$ and maps induced by pre- and postcomposition). It is a complex vector space by ${\mathrm{Hom}}_G(V,W)$ is a $k$-vector space and ${\mathrm{End}}_G(V)$ is a $k$-algebra and the group-action/group-ring-module correspondence. It is finite-dimensional: each Specht module is spanned by polytabloids indexed by the finite set of tableaux of its shape (Tableaux and standard tableaux, Column antisymmetrizers, polytabloids, and Specht modules), and $K^{\lambda \mathrm{/}\mu }$ is a linear subspace of the finite-dimensional space of complex-linear maps from $S^{\mu }$ to $V_{m,r}^{\lambda }$.

Give it an $S_r$-action by postcomposition on the target:

$(\tau \cdot \phi )(v):={\iota }_{m,r}(1,\tau )\cdot \phi (v)(\tau \in S_r,\phi \in K^{\lambda \mathrm{/}\mu },v\in S^{\mu })\mathrm{.}$

This action is well defined. For $\sigma \in S_m$, the two elements ${\iota }_{m,r}(\sigma ,1)$ and ${\iota }_{m,r}(1,\tau )$ commute in $S_m\times S_r$, so

$(\tau \cdot \phi )(\sigma \cdot v)={\iota }_{m,r}(1,\tau ){\iota }_{m,r}(\sigma ,1)\cdot \phi (v)={\iota }_{m,r}(\sigma ,1){\iota }_{m,r}(1,\tau )\cdot \phi (v)=\sigma \cdot (\tau \cdot \phi )(v)\mathrm{.}$

Thus $\tau \cdot \phi$ remains $\mathbb{C}[S_m]$-linear, and componentwise multiplication shows $({\tau }_1{\tau }_2)\cdot \phi ={\tau }_1\cdot ({\tau }_2\cdot \phi )$ with the identity acting trivially. This makes $K^{\lambda \mathrm{/}\mu }$ a finite-dimensional complex $S_r$-module. The construction includes $m=0$ and $r=0$. It defines the skew object as an intertwiner space over $\mathbb{C}$; no skew polytabloid filtration over an arbitrary field is asserted. No choice principle is used.