This question concerns the characteristic $0$ representation theory of the symmetric group $S\_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way.


First, a bit of background. The finite-dimensional irreducible representations of $S\_n$ are given by the [Specht modules](https://en.wikipedia.org/wiki/Specht_modules) $S^{\mu}$. Here $\mu$ is a partition of $n$, which is best visualized as a [Young diagram](https://en.wikipedia.org/wiki/Young_diagram). There are classical rules for restricting $S^{\mu}$ to $S\_{n-1}$ and inducing $S^{\mu}$ to $S\_{n+1}$ (these are known as branching rules). Namely, we have the following.


1. The restriction of $S^{\mu}$ to $S\_{n-1}$ is isomorphic to the direct sum of the $S\_{n-1}$-representations $S^{\mu'}$ as $\mu'$ goes over all ways of removing a box from the Young diagram for $\mu$ (while staying in the world of Young diagrams).
2. The induction of $S^{\mu}$ to $S\_{n+1}$ is isomorphic to the direct sum of the $S\_{n+1}$-representations $S^{\mu'}$ as $\mu'$ goes over all ways of adding a box to the Young diagram for $\mu$ (again while staying in the world of Young diagrams).


These two rules are equivalent by [Frobenius reciprocity](https://en.wikipedia.org/wiki/Frobenius_reciprocity).


There is a nice concrete proof of the restriction rule (I believe that it is due to Peel, though I first learned about it from James's book "The representation theory of the symmetric groups"). Assume that the rows of the Young diagram for $\mu$ from which we can remove a box are $r\_1 < \ldots < r\_k$, and denote by $\mu\_i$ the partition of $n-1$ obtained by removing a box from the $r\_i^{\text{th}}$ row of $\mu$. There is then a sequence
$$0=V\_0 \subset V\_1 \subset \cdots \subset V\_k = S^{\mu}$$
of $S\_{n-1}$-modules such that $V\_i/V\_{i-1} \cong S^{\mu\_i}$. In fact, recalling that $S^{\mu}$ is generated by elements corresponding to Young tableaux of shape $\mu$ (known as polytabloids), the module $V\_i$ is the subspace spanned by the polytabloids in which $n$ appears in a row between $1$ and $i$.


**Question** : Is there a similarly concrete proof of the induction rule (in particular, a proof which does not appeal to Frobenius reciprocity and the restriction rule)?



Making the branching rule for the symmetric group concrete
Chapter 17 of James' book, The Representation Theory of the Symmetric Groups, is about modules which have Specht filtrations (where the field is arbitrary) and includes induction of the Specht modules as a special case. (See in particular Example 17.16.) It gives a concrete proof of the branching rule for induction without using restriction. 


