TITLE: Stabilization of representation of the symmetric group
QUESTION [13 upvotes]: In their seminal work on "representation stability" (https://arxiv.org/abs/1008.1368), Church and Farb deal with a stabilization procedure for representations (up to isomorphism) (over $\mathbf C$) of the symmetric group. It's a stabilization in the sense that it produces a representation of $\mathfrak S_{n+1}$ out of a representation of $\mathfrak S_n$.
The procedure is defined for irreducible representations as follows.
Irreducible representations are in bijective correspondence with Young diagrams and their procedure consists in sending the representation associated to a Young diagram  on the irreducible representation associated to the Young diagram with one more box on the first line (the biggest one).
In general, a representation is isomorphic to a sum of irreducibles and its image is the sum of the images of each summand.
My question is: does this stabilization procedure correspond to an intrinsic procedure on $\mathfrak S_n$ representations? Or equivalently, can you describe it only in terms of representations (without knowing the diagrams of the irreducible summands and/or in a functorial way)?

REPLY [5 votes]: Here is one idea that is implicit in the $\rm FI$ module approach:
If you have a fixed $S_{n_0}$  representation $W$,  then the sequence of representations  $n \mapsto {\rm Ind}_{S_{n_0} \times S_{n - n_0}}^{S_n}  W$,  for $n \geq n_0$ is representation stable:  by the Pieri rule, the decomposition into irreducibles eventually only differs from $n$ to $n+1$ by adding a box to  each top row.
And,  if a sequence of $S_n$ representations  $n \mapsto V_n$  is representation stable, it can be written as quotient of families of $S_n$ representations that are induced from a finite level.  More precisely, there exist finitely many $S_{n_i}$  representations  $W_i$  and $S_{m_j}$ representations $U_j$ and and a family of maps $$f(n): \bigoplus_j {\rm Ind}_{S_{n_j} \times S_{n - n_j}}^{S_n} U_j \to  \bigoplus_i  {\rm Ind}_{S_{n_i} \times S_{n - n_i}}^{S_n} W_i   $$
such that $V_n$ is the cokernel of $f(n)$ for $n\gg 0$.  You can see this from the Pieri rule as well.  
So this doesn't give an answer to what "adding a box" means intrinsically for passing from $S_n$ to $S_{n+1}$ representations.  Instead, it suggests that eventually differing by "adding a box" should be equivalent to being a quotient of representations  that are induced from some finite set of  $S_{n_i}$ such that the complement of $n_i$ acts trivially.  This isn't true in the setup I've given so far, because there should be some restriction on the maps $f(n)$ that makes them uniform in $n$. This is what $\rm FI$ modules formalize:  the domain and target of the maps $f(n)$ are exactly "free" or projective $\rm FI$ modules, and the uniformity condition is that the $f(n)$ be a map of $\rm FI$ modules.