The outer Littlewood–Richardson rule

For each $t$, the label shift ${\beta }_t(i)=i+1$ (empty if $t=0$) gives the group isomorphism $c_t(\sigma )={\beta }_t\sigma {\beta }_t^{-1}$ from zero-based to one-based permutations. It preserves products, cycle types and the ordered block embeddings. For a one-based subgroup $H^1$ and module $W$, put $H^0=c_{m+n}^{-1}(H^1)$ and $h{\cdot }_{0}w=c_{m+n}(h){\cdot }_{1}w$. Pullback of induced functions is $F^0(g)=F^1(c_{m+n}(g))$, with inverse composition by $c_{m+n}^{-1}$; it satisfies $F^0(gh)=h^{-1}{\cdot }_0{F}^0(g)$ and intertwines left translation because $c_{m+n}$ preserves products. Relabeling tableaux by the same shift identifies tabloids, conjugates their column stabilizers and preserves signs, hence identifies the Specht actions in [F14]. Cycle-type preservation leaves [F6] unchanged. Thus the character and induction formulas use compatible group conventions.

By [F7] and [F8], $\mathrm{ch}({\chi }^{\mu }\circ {\chi }^{\nu })=\mathrm{ch}({\chi }^{\mu })\mathrm{ch}({\chi }^{\nu })=s_{\mu }{s}_{\nu }$. Applying the Schur expansion [F9] and the linearity of $\mathrm{ch}$ in [F6] gives $\mathrm{ch}({\chi }^{\mu }\circ {\chi }^{\nu })={\sum }_{\lambda ⊢m+n}{c}_{\mu \nu }^{\lambda }\mathrm{ch}({\chi }^{\lambda })=\mathrm{ch}\left({\sum }_{\lambda ⊢m+n}{c}_{\mu \nu }^{\lambda }{\chi }^{\lambda }\right)$. The sum is finite by [F9].

The two class functions inside $\mathrm{ch}$ in step 1.2 have the same characteristic. Injectivity [F10] therefore gives ${\chi }^{\mu }\circ {\chi }^{\nu }={\sum }_{\lambda ⊢m+n}{c}_{\mu \nu }^{\lambda }{\chi }^{\lambda }$ as class functions on $S_{m+n}$.

Let $V={\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}(S^{\mu }⊠S^{\nu })$ and $W={⨁}_{\lambda ⊢m+n}(S^{\lambda }{)}^{\oplus c_{\mu \nu }^{\lambda }}$. These are finite-dimensional complex representations: each Specht module is spanned by finitely many polytabloids by [F14], their external tensor product is finite-dimensional by [F15], the sum has finite support by [F9], and the covariant induction space [F4] is a subspace of the finite-dimensional function space from the finite group $S_{m+n}$ to $S^{\mu }\otimes S^{\nu }$. The character of $V$ is ${\chi }^{\mu }\circ {\chi }^{\nu }$ by [F3, F5]; the character of $W$ is ${\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{\chi }^{\lambda }$ by additivity [F16]. Step 2.1 makes these characters equal, so [F11] gives $V\cong W$.

By [F12, F13], the summands $S^{\lambda }$ in $W$ are pairwise inequivalent irreducible modules, and the direct sum in step 3.1 contains exactly $c_{\mu \nu }^{\lambda }$ copies of each one. Thus this is the multiplicity of $S^{\lambda }$ in the induced module as well. The conclusion uses the outer induction product defined in [F3], not a tensor product of two modules for the same symmetric group; every relabeling and sum is explicit and finite, so no form of the axiom of choice is used.