The Littlewood–Richardson coefficients are the multiplicities $$ c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu)) $$ and the Littlewood–Richardson rule says that these coefficients are enumerated by the semistandard Young tableaux satisfying the lattice permutation condition. Moreover, using James' approach one can actually construct the homomorphisms from these tableaux (so we can work on a structural level). There are also plenty of proofs using only combinatorics (of Schur functions).

Is there a proof on a level between these two? The characters of the symmetric groups have "more structure" than simple combinatorics but have less structure than representations. Is there an entirely character-theoretic proof of this rule? I imagine that any such proof would use the Murnaghan–Nakayama rule for (skew representations of) symmetric groups.

P.S. James gave one of the first proofs of the Littlewood–Richardson rule in 1977 (funnily enough, this wasn't proven by Littlewood and Richardson). James' proof calculates these homomorphisms over any field (whereas his contemporaries Schützenberger (1977) and Thomas (1974) only considered the semisimple case using combinatorics) and so his work is far more impressive and benefits from being self-contained. Yet he gets little recognition of this (see Wikipedia for example). His proof can be found here https://doi.org/10.1016/0021-8693(77)90380-5