$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction yields an irreducible representation $V_\lambda \boxtimes V_\mu$ of $S_{d_1} \times S_{d_2} \subset S_d$.

In their book on representation theory, Fulton and Harris define the Littlewood–Richardson numbers $N_{\lambda\mu\nu}$ as being the ones such that $S_\lambda S_\mu = \sum_{\nu} N_{\lambda\mu\nu} S_\nu$, where $S_\lambda$ denotes the Schur polynomial and the sum is over partitions of $d$. They go on to claim in (4.41) that

$$\Ind_{S_{d_1} \times S_{d_2}}^{S_d}(V_\lambda \boxtimes V_\mu) = \bigoplus_{\nu} V_{\nu}^{\oplus N_{\lambda \mu \nu}},$$

where $V_{\nu}$ of course denotes the irreducible representation of $S_d$ coming from $\nu$.

I agree completely that (4.41) follows from what Fulton and Harris take to be the definition of the Littlewood–Richardson numbers, combined with other things they wrote in lecture 4 and appendix A of their book — it is a consequence of the fact that the Frobenius characteristic map is a ring homomorphism that takes characters of irreducible representations to Schur polynomials, which is essentially established earlier in the lecture.

However, their justification is totally different from what I had in mind: they write (with some of their notation replaced to make it consistent with mine) "Indeed, by the exercise, the character of $\Ind_{S_{d_1} \times S_{d_2}}^{S_d}(V_\lambda \boxtimes V_\mu)$ is the product of the corresponding determinants, and, by the definition of the Littlewood–Richardson numbers, that is the sum of the characters $N_{\lambda \mu \nu}V_{\nu}$". Here, "the exercise" seems to refer to Exercise 4.40, which provides a formula for the character of $V_\nu$ as a linear combination of characters of inductions of the trivial representation of various Young subgroups of $S_d$. I did that exercise — you can do it without writing down the Frobenius characteristic map by just arguing about how the recurrences involving Kostka numbers for representations and for symmetric functions are the same and therefore the Jacobi–Trudi identity also holds for characters. But I really don't see how the determinantal formula of 4.40 helps at all in the proof of (4.41). Since it seems like Fulton and Harris employ kind of an interesting viewpoint in service of completely avoiding actually writing down the Frobenius characteristic map, I thought I would ask if anyone has clarifying remarks on the full detail of this justification, especially given that it might shed light on the usefulness of Exercise 4.40.