Conjugation and exchange symmetries of the Littlewood–Richardson coefficients

By [F6], $s_{\mu }{s}_{\nu }=s_{\nu }{s}_{\mu }$. Expanding each side by [F2] and comparing coefficients in the Schur basis [F4] gives $c_{\mu \nu }^{\lambda }=c_{\nu \mu }^{\lambda }$ for every $\lambda$.

Apply the ring homomorphism $\omega$ [F3] to the expansion in [F2]. Since $\omega (s_{\eta })=s_{{\eta }^{\mathrm{\prime }}}$, this gives $s_{{\mu }^{\mathrm{\prime }}}{s}_{{\nu }^{\mathrm{\prime }}}={\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{s}_{{\lambda }^{\mathrm{\prime }}}={\sum }_{\eta }{c}_{\mu \nu }^{{\eta }^{\mathrm{\prime }}}{s}_{\eta }$, where the reindexing $\eta ={\lambda }^{\mathrm{\prime }}$ is valid by [F5]. The LR expansion [F2] applied to ${\mu }^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }}$ also gives $s_{{\mu }^{\mathrm{\prime }}}{s}_{{\nu }^{\mathrm{\prime }}}={\sum }_{\eta }{c}_{{\mu }^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}}^{\eta }{s}_{\eta }$. Uniqueness in [F4] implies $c_{{\mu }^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}}^{\eta }=c_{\mu \nu }^{{\eta }^{\mathrm{\prime }}}$; taking $\eta ={\lambda }^{\mathrm{\prime }}$ and using ${\lambda }^{\mathrm{\prime }\mathrm{\prime }}=\lambda$ proves $c_{\mu \nu }^{\lambda }=c_{{\mu }^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}}^{{\lambda }^{\mathrm{\prime }}}$.

If a partition is empty, the LR expansion [F2] includes the unit and all other coefficients are zero by [F1], so both symmetries still hold. If a coefficient is zero because its containment or size condition fails, [F1] gives zero and conjugation preserves those conditions by [F5]; when the LR set is empty despite valid containment and size, step 1.2 already proves the conjugate coefficient is equal to it. Thus the zero, unit, and boundary cases, including $\lambda =\mu$, require no strict-containment assumption. All coefficient comparisons are in finite homogeneous degrees. No tableau representatives, bases, or other objects are chosen, and no axiom of choice is used.