Multiplicity-free complex Specht induction

Put $m:=n+1\ge 1$, $G:=S_{n+1}$, $H:=S_n$ and $W:=S_{\mathbb{C}}^{\lambda }$; by [F1] the induced module ${\mathrm{Ind}}_H^{G}W$ is a finite-dimensional complex $G$-module, and by [F9] for every $\nu ⊢m$ the modules $S_{\mathbb{C}}^{\nu }$ and, for every $x$, $S_{\mathbb{C}}^{\nu -x}$ are finite-dimensional complex representations of $G$ and of $H$ respectively.

The right-hand module ${⨁}_{y\in \mathrm{Add}(\lambda )}{S}_{\mathbb{C}}^{\lambda +y}$ is a finite direct sum of irreducible $G$-modules with multiplicity exactly $1$ at those $\nu$ of the form $\nu =\lambda +y$ and multiplicity $0$ at all other $\nu$: the addable nodes $y$ of $[\lambda ]$ give pairwise distinct partitions $\lambda +y$ and hence pairwise non-isomorphic summands by [F4] and [F8].

For every $\nu ⊢m$, the adjunction [F2] with $R=\mathbb{C}$ gives a $\mathbb{C}$-linear isomorphism ${\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,S_{\mathbb{C}}^{\nu })\cong {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^G{S}_{\mathbb{C}}^{\nu })$.

For every $\nu ⊢m$ the restriction rule [F3] applies with its parameter equal to $m\ge 1$, so ${\mathrm{Res}}_H^G{S}_{\mathbb{C}}^{\nu }\cong {⨁}_{x\in \mathrm{Rem}(\nu )}{S}_{\mathbb{C}}^{\nu -x}$; composing with this isomorphism and splitting a homomorphism into a direct sum into its finitely many components gives ${\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^G{S}_{\mathbb{C}}^{\nu })\cong {⨁}_{x\in \mathrm{Rem}(\nu )}{\mathrm{Hom}}_{S_n}(S_{\mathbb{C}}^{\lambda },S_{\mathbb{C}}^{\nu -x})$.

For each $x\in \mathrm{Rem}(\nu )$ the summand ${\mathrm{Hom}}_{S_n}(S_{\mathbb{C}}^{\lambda },S_{\mathbb{C}}^{\nu -x})$ is one-dimensional when $\nu -x=\lambda$ and is zero otherwise: both arguments are irreducible complex $S_n$-modules and the partitions $\lambda$ and $\nu -x$ of $n$ are either equal or distinct, so [F5] applies.

By [F6] the module ${\mathrm{Ind}}_H^{G}W$ is completely reducible, so it is isomorphic to a finite direct sum ${⨁}_{\nu ⊢m}(S_{\mathbb{C}}^{\nu }{)}^{\oplus m_{\nu }}$ for nonnegative integers $m_{\nu }$; [F4] makes the indexing complete and irredundant, and [F5] together with additivity of $\mathrm{Hom}$ in each argument gives $m_{\nu }={\dim }_{\mathbb{C}}{\mathrm{Hom}}_G(S_{\mathbb{C}}^{\nu },{\mathrm{Ind}}_H^{G}W)={\dim }_{\mathbb{C}}{\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,S_{\mathbb{C}}^{\nu })$.

Steps 2.1, 2.2 and 2.3 combine to ${\dim }_{\mathbb{C}}{\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,S_{\mathbb{C}}^{\nu })=\mathrm{\#}\{x\in \mathrm{Rem}(\nu ):\nu -x=\lambda \}$.

The set $\{x\in \mathrm{Rem}(\nu ):\nu -x=\lambda \}$ is in bijection with $\{y\in \mathrm{Add}(\lambda ):\lambda +y=\nu \}$ by the identity map on nodes: if $x$ is removable with $[\nu ]\setminus \{x\}=[\lambda ]$, then $y:=x$ is a point outside $[\lambda ]$ with $[\lambda ]\cup \{y\}=[\nu ]$ a Young diagram, so $y$ is addable for $\lambda$ and $\lambda +y=\nu$; conversely if $y$ is addable with $[\lambda ]\cup \{y\}=[\nu ]$, then $x:=y$ lies in $[\nu ]$ with $[\nu ]\setminus \{x\}=[\lambda ]$ a Young diagram, so $x$ is removable for $\nu$ and $\nu -x=\lambda$. Hence by step 3.1 the dimension ${\dim }_{\mathbb{C}}{\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,S_{\mathbb{C}}^{\nu })$ equals $1$ if $\nu =\lambda +y$ for some addable node $y$ of $[\lambda ]$, and equals $0$ otherwise.

By step 2.4 and step 4.1 the multiplicities of ${\mathrm{Ind}}_H^{G}W$ are $1$ exactly at the partitions $\nu =\lambda +y$ with $y$ addable for $\lambda$ and $0$ at all other $\nu ⊢m$; by step 1.2 the module ${⨁}_y{S}_{\mathbb{C}}^{\lambda +y}$ has the same multiplicities, and both modules are completely reducible by [F6]. Grouping each module into its isotypic components, which by [F7] are determined by the multiplicities alone, gives the asserted isomorphism ${\mathrm{Ind}}_{S_n}^{S_{n+1}}{S}_{\mathbb{C}}^{\lambda }\cong {⨁}_{y\in \mathrm{Add}(\lambda )}{S}_{\mathbb{C}}^{\lambda +y}$ with each summand occurring once.

Boundary and consistency check. For $n=0$ one has $\lambda =\mathrm{\varnothing }$, $H=S_0=\{1\}$ and $m=1$; the single partition $\nu =(1)$ has the single removable node $x=(1,1)$ with $\nu -x=\mathrm{\varnothing }=\lambda$, while $\mathrm{Add}(\mathrm{\varnothing })=\{(1,1)\}$ by [F8], so step 5.1 gives ${\mathrm{Ind}}_{S_0}^{S_1}{S}_{\mathbb{C}}^{\mathrm{\varnothing }}\cong S_{\mathbb{C}}^{(1)}$; every partition $\lambda ⊢n$ with $n\ge 1$ has at least the addable node opening a new row, so the displayed direct sum is never empty in that case, and the theorem uses the finite groups $S_n$, $S_{n+1}$ and finitely many partitions throughout, invoking no choice principle. This proves the Statement.