The characteristic of a Specht character is a Schur function

For every $\mu ⊢n$, taking characters in Young's rule [F1] and using additivity on direct sums [F3] gives ${\phi }^{\mu }={\sum }_{\lambda ⊢n}{K}_{\lambda \mu }{\chi }^{\lambda }$ in $\mathrm{c}\mathrm{f}(S_n)$, the sum being finite.

Applying the $\mathbb{Z}$-linear map $\mathrm{ch}$ to step 1.1 and using [F5] gives $h_{\mu }=\mathrm{ch}({\phi }^{\mu })={\sum }_{\lambda ⊢n}{K}_{\lambda \mu }\mathrm{ch}({\chi }^{\lambda })$ in ${\mathrm{\Lambda }}_{\mathbb{C}}^n$.

Subtracting the identity $h_{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }{s}_{\lambda }$ of [F6] from step 2.1 yields ${\sum }_{\lambda ⊢n}{K}_{\lambda \mu }(\mathrm{ch}({\chi }^{\lambda })-s_{\lambda })=0$ for every $\mu ⊢n$, a homogeneous linear system with coefficient matrix $K^{\mathsf{T}}$, where $K=(K_{\lambda \mu })$; since $K$ is unitriangular in a linear extension of dominance, both $K$ and $K^{\mathsf{T}}$ are invertible over $\mathbb{Z}$, so the only solution is the zero vector and $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ for every $\lambda ⊢n$.

Inverting the integral matrices, $s_{\lambda }={\sum }_{\mu ⊢n}(K^{-1}{)}_{\mu \lambda }{h}_{\mu }$ with $(K^{-1}{)}_{\mu \lambda }\in \mathbb{Z}$; substituting $h_{\mu }={\sum }_{\rho ⊢n}N(\mu ,\rho )p_{\rho }\mathrm{/}{z}_{\rho }$ with integral $N(\mu ,\rho )$ gives $s_{\lambda }={\sum }_{\rho ⊢n}{c}_{\lambda \rho }{p}_{\rho }\mathrm{/}{z}_{\rho }$ with $c_{\lambda \rho }={\sum }_{\mu }(K^{-1}{)}_{\mu \lambda }N(\mu ,\rho )\in \mathbb{Z}$. Since $\{p_{\rho }\mathrm{/}{z}_{\rho }:\rho ⊢n\}$ is a $\mathbb{Q}$-basis of ${\mathrm{\Lambda }}_{\mathbb{Q}}^n$ and $\mathrm{ch}({\chi }^{\lambda })={\sum }_{\rho }{\chi }^{\lambda }(\rho )p_{\rho }\mathrm{/}{z}_{\rho }$ by definition, comparing coefficients gives ${\chi }^{\lambda }(\rho )=c_{\lambda \rho }\in \mathbb{Z}$ for every $\rho ⊢n$.

The characters ${\chi }^{\lambda }$ are pairwise distinct irreducible characters of $S_n$ and the irreducible characters are $\mathbb{Z}$-linearly independent [F9]; since $R(S_n)$ is by definition their integral span [F4], the family $\{{\chi }^{\lambda }:\lambda ⊢n\}$ is a $\mathbb{Z}$-basis of $R(S_n)$. By [F7] the family $\{s_{\lambda }:\lambda ⊢n\}$ is a $\mathbb{Z}$-basis of ${\mathrm{\Lambda }}^n$, and by step 3.1 the map $\mathrm{ch}$ carries the first basis bijectively onto the second; step 4.1 shows that all character values are integers.