The Littlewood--Richardson tensor-product rule

Decomposition. By [F2] the multiplicity of $S_{\nu }(V)$ in $S_{\lambda }(V)\otimes S_{\mu }(V)$ equals $c_{\lambda \mu }^{\nu }$ for every partition $\nu$ with $\mathrm{\ell }(\nu )\le r$. The module is completely reducible by the tensor-power retraction proved in the supplier of [F2], and its irreducible summands are among the pairwise non-isomorphic simple modules $S_{\nu }(V)$ with $\mathrm{\ell }(\nu )\le r$ by [F1]; therefore $S_{\lambda }(V)\otimes S_{\mu }(V)\cong {⨁}_{\nu :\mathrm{\ell }(\nu )\le r}{S}_{\nu }(V{)}^{\oplus c_{\lambda \mu }^{\nu }},$ where the sum is finite by [F4] and the vanishing statement of [F3] removes all $\nu$ with $\lambda \subseteq ̸\nu$ or $\mathrm{\mid }\nu \mathrm{\mid }\mathrm{\ne }\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$.

Characters. Taking characters in step 1.1 and using additivity and multiplicativity of the character together with $\mathrm{ch}{S}_{\nu }(V)=s_{\nu }$ [F1] gives $s_{\lambda }{s}_{\mu }={\sum }_{\nu :\mathrm{\ell }(\nu )\le r}{c}_{\lambda \mu }^{\nu }{s}_{\nu }(x_1,\dots ,x_r)$, where terms with $\mathrm{\ell }(\nu )>r$ are $0$ by definition of $s_{\nu }(x_1,\dots ,x_r)$ and $S_{\nu }(V)=0$.

Independence of $r$. The coefficient of $s_{\nu }$ in step 2.1 is $c_{\lambda \mu }^{\nu }$, the number of LR tableaux of shape $\nu \mathrm{/}\lambda$ and content $\mu$ (Littlewood--Richardson tableaux and coefficients); this is a count of tableaux of a fixed skew shape and content, so it does not mention the rank $r$ at all, and the multiplicity statement of step 1.1 identifies the same integer as the multiplicity in the tensor product for every $r$ with $\mathrm{\ell }(\lambda ),\mathrm{\ell }(\mu )\le r$. Hence the coefficients $c_{\lambda \mu }^{\nu }$ appearing in the decomposition are independent of $r$, as claimed.