The skew multiplicity module decomposes with Littlewood–Richardson multiplicities over $\mathbb{C}$

The module $K^{\lambda \mathrm{/}\mu }$ is finite-dimensional by [F1], so Maschke's theorem [F2] gives a finite decomposition into irreducibles. By the complete irredundant classification [F3], write $K^{\lambda \mathrm{/}\mu }\cong {⨁}_{\nu ⊢r}(S^{\nu }{)}^{\oplus m_{\nu }}$. The character multiplicity formula [F8] gives $m_{\nu }=⟨{\chi }_K,{\chi }^{\nu }⟩$, and the intertwiner-dimension formula [F7] makes this $\dim {\mathrm{Hom}}_{S_r}(S^{\nu },K^{\lambda \mathrm{/}\mu })$.

Fix $\nu ⊢r$ and write $V=V_{m,r}^{\lambda }$. The definition [F1] identifies ${\mathrm{Hom}}_{S_r}(S^{\nu },K^{\lambda \mathrm{/}\mu })$ with ${\mathrm{Hom}}_{S_r}(S^{\nu },{\mathrm{Hom}}_{S_m}(S^{\mu },V))$. Under the $\mathbb{C}$-linear currying isomorphism [F6], a map $f$ corresponds to $g:S^{\nu }{\otimes }_{\mathbb{C}}{S}^{\mu }\to V$, $g(y\otimes u)=f(y)(u)$. The condition that each $f(y)$ is $S_m$-linear is $g(y\otimes \sigma u)={\iota }_{m,r}(\sigma ,1)g(y\otimes u)$; the $S_r$-equivariance of $f$ is $g(\tau y\otimes u)={\iota }_{m,r}(1,\tau )g(y\otimes u)$. Since the two actions commute by [F1], these are exactly the equivariance conditions for the product group after flipping $y\otimes u$ to $u\otimes y$. By [F5] the flipped source is $S^{\mu }⊠S^{\nu }$, so currying restricts to an isomorphism ${\mathrm{Hom}}_{S_r}(S^{\nu },K^{\lambda \mathrm{/}\mu })\cong {\mathrm{Hom}}_{S_m\times S_r}(S^{\mu }⊠S^{\nu },V)$.

The character $\theta ={\chi }^{\mu }⊠{\chi }^{\nu }$ is honest by [F3, F5] and has norm one by [F4]. Maschke's theorem [F2] gives a finite decomposition $\theta ={\sum }_i{n}_i{\eta }_i$ into irreducible characters. By the multiplicity formula [F8], $n_i=⟨\theta ,{\eta }_i⟩$; therefore $1=⟨\theta ,\theta ⟩={\sum }_i{n}_i⟨\theta ,{\eta }_i⟩={\sum }_i{n}_i^2$, so exactly one constituent occurs once and $S^{\mu }⊠S^{\nu }$ is irreducible. Apply [F7] to the Hom space in step 1.2: its dimension is $⟨{\chi }_V,{\chi }^{\mu }⊠{\chi }^{\nu }{⟩}_{S_m\times S_r}$. The restriction formula [F9] expands ${\chi }_V$ in the orthonormal external-product basis [F4], with coefficient $c_{\mu \nu }^{\lambda }$ on this term. Thus $\dim {\mathrm{Hom}}_{S_r}(S^{\nu },K^{\lambda \mathrm{/}\mu })=c_{\mu \nu }^{\lambda }$.

Comparing steps 1.1 and 2.1 gives $m_{\nu }=c_{\mu \nu }^{\lambda }$ for every $\nu ⊢r$, which proves the displayed $S_r$-module decomposition. A zero coefficient means the corresponding irreducible does not occur; a coefficient one gives exactly one copy.

Additivity of the Frobenius characteristic and [F10] give $\mathrm{ch}(K^{\lambda \mathrm{/}\mu })={\sum }_{\nu ⊢r}{m}_{\nu }{s}_{\nu }$; substituting step 3.1 and using the skew expansion [F11] yields $\mathrm{ch}(K^{\lambda \mathrm{/}\mu })=s_{\lambda \mathrm{/}\mu }$. If $\mu =\lambda =\mathrm{\varnothing }$, then $m=r=0$ and [F1, F12] give $K={\mathrm{Hom}}_{\mathbb{C}}(\mathbb{C},\mathbb{C})=\mathbb{C}=S^{\mathrm{\varnothing }}$, with coefficient one. At the endpoints, if $m=0$, evaluation at $1\in S^{\mathrm{\varnothing }}$ identifies $K$ with $S^{\lambda }$ as an $S_r$-module; if $r=0$, then $\mu =\lambda$ and $K={\mathrm{End}}_{S_m}(S^{\lambda })$ is one-dimensional by steps 1.1 and 2.1, matching the empty-factor coefficient. These cases also show the result at degree zero. All direct sums are finite, indexed by partitions of $r$, and use Maschke's finite-group decomposition; no choice principle is used.