Restriction to the two-block Young subgroup has LR multiplicities

For $\nu ⊢n-r$ and $\mu ⊢r$ put ${\theta }_{\nu \mu }:={\chi }^{\nu }⊠{\chi }^{\mu }\in R(K)$; by [F2] it is an honest character of $K$, and the family $({\theta }_{\nu \mu }{)}_{\nu ,\mu }$ is orthonormal and $\mathbb{Z}$-spans $R(K)$. By [F5] its induction to $S_n$ is the outer product ${\chi }^{\nu }\circ {\chi }^{\mu }={\sum }_{\lambda ⊢n}{c}_{\nu \mu }^{\lambda }{\chi }^{\lambda }$.

Fix $\lambda ⊢n$. Applying [F1] and [F4] to the character ${\theta }_{\nu \mu }$ of $K$ and the character ${\chi }^{\lambda }$ of $S_n$, and then using [F5] and the orthonormality of the irreducible characters ${\chi }^{{\lambda }^{\mathrm{\prime }}}$ from [F3], gives $⟨{\theta }_{\nu \mu },{\chi }^{\lambda }{\mathrm{\mid }}_K{⟩}_K=⟨{\mathrm{Ind}}_{S_{n-r}\times S_r}^{S_n}{\theta }_{\nu \mu },{\chi }^{\lambda }{⟩}_{S_n}={\sum }_{{\lambda }^{\mathrm{\prime }}⊢n}{c}_{\nu \mu }^{{\lambda }^{\mathrm{\prime }}}⟨{\chi }^{{\lambda }^{\mathrm{\prime }}},{\chi }^{\lambda }⟩=c_{\nu \mu }^{\lambda }$.

The characters ${\theta }_{\nu \mu }$ are exactly the irreducible characters of $K$. Indeed, every irreducible character ${\chi }_i$ of $K$ lies in $R(K)$ by [F3] and hence is a finite integral combination ${\chi }_i={\sum }_{\nu ,\mu }{n}_{\nu \mu }{\theta }_{\nu \mu }$ because the ${\theta }_{\nu \mu }$ form a $\mathbb{Z}$-basis of $R(K)$ by [F2]. Computing the norm with [F3] and the orthonormality of the ${\theta }_{\nu \mu }$ gives $1=⟨{\chi }_i,{\chi }_i⟩={\sum }_{\nu ,\mu }{n}_{\nu \mu }^2$, so exactly one coefficient equals $\pm 1$ and the others are $0$, that is, ${\chi }_i=\pm {\theta }_{\nu \mu }$ for a single pair $(\nu ,\mu )$; a character has positive degree, so ${\chi }_i={\theta }_{\nu \mu }$. Hence the irreducible characters of $K$ form a subfamily of the orthonormal family $({\theta }_{\nu \mu })$; the irreducibles are an orthonormal basis of the class functions by [F3], so their number is $\dim \mathrm{c}\mathrm{f}(K)$, while an orthonormal family in $\mathrm{c}\mathrm{f}(K)$ has at most $\dim \mathrm{c}\mathrm{f}(K)$ elements; the numbers are therefore equal and the two families coincide.

The restricted character ${\chi }^{\lambda }{\mathrm{\mid }}_K$ is a class function on $K$, and by [F3] it expands in the orthonormal basis of irreducible characters, which is the family $({\theta }_{\nu \mu })$ by step 2.2, with coefficient $⟨{\chi }^{\lambda }{\mathrm{\mid }}_K,{\theta }_{\nu \mu }{⟩}_K$. This is the conjugate of $⟨{\theta }_{\nu \mu },{\chi }^{\lambda }{\mathrm{\mid }}_K{⟩}_K=c_{\nu \mu }^{\lambda }$ from step 2.1 and hence equals that same real integer: ${\chi }^{\lambda }{\mathrm{\mid }}_K={\sum }_{\nu ⊢n-r}{\sum }_{\mu ⊢r}⟨{\theta }_{\nu \mu },{\chi }^{\lambda }{\mathrm{\mid }}_K{⟩}_K{\theta }_{\nu \mu }$. Substituting the value $c_{\nu \mu }^{\lambda }$ from step 2.1 into each coefficient gives the displayed identity of the statement.

The coefficient of the irreducible character ${\theta }_{\nu \mu }={\chi }^{\nu }⊠{\chi }^{\mu }$ in the expansion of step 3.1 is $c_{\nu \mu }^{\lambda }$, so this is the multiplicity of ${\chi }^{\nu }⊠{\chi }^{\mu }$ in ${\chi }^{\lambda }{\mathrm{\mid }}_K$. If $r=n$ then $K=S_n$ and the only partition of $n-r=0$ is $\nu =\mathrm{\varnothing }$, so the identity reads ${\chi }^{\lambda }={\sum }_{\mu ⊢n}{c}_{\mathrm{\varnothing }\mu }^{\lambda }{\chi }^{\mu }$; comparing with the orthonormal-basis expansion ${\chi }^{\lambda }={\sum }_{\mu }{\delta }_{\lambda \mu }{\chi }^{\mu }$ from [F3] gives $c_{\mathrm{\varnothing }\mu }^{\lambda }={\delta }_{\lambda \mu }$, so the multiplicity is $1$ for $\mu =\lambda$ and $0$ otherwise. Similarly $r=0$ gives $\mu =\mathrm{\varnothing }$ and $c_{\nu \mathrm{\varnothing }}^{\lambda }={\delta }_{\lambda \nu }$; when $n=0$ both partitions are empty and the coefficient is $1$.