Stabilization of representation of the symmetric group

There are two conceptual approaches to what Church-Farb did: the theory of "central stability" that I defined in my paper

A. Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), no. 3, 987-1027.

and Church-Ellenberg-Farb's notion of FI-modules from their paper

T. Church, J. Ellenberg, and B. Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910

The point of both of these papers is that the "add a box to the upper right hand corner" operation that Church-Farb observed is not a good description of the actual pattern. After all, if $V_{n+1}$ is obtained from stabilizing $V_n$ via C-F's recipe, then C-F does not give you a $S_n$-equivariant map from $V_n$ to $V_{n+1}$ (which exists in all the examples they care about). And in these examples those map are not the ones you might guess -- if $W$ is an $S_n$-subrepresentation of $V_n$, then the image of $W$ in $V_{n+1}$ does not lie in a single $S_{n+1}$-subrepresentation.

Rather, both of these papers give more primitive stabilization operations and then prove as a theorem that if you keep performing those stabilization operations, then eventually your representations will settle down into C-F's original pattern.

It turns out that all the examples C-F gave form both finitely generated FI-modules and also satisfy central stability. What is more, a sequence of symmetric group representations (satisfying some simple conditions to ensure that this make sense) turn out to satisfy central stability if and only if they form a finitely generated FI-module. It is immediate that central stability implies that the resulting FI-module is finitely generated; the other direction is proved by Church-Ellenberg-Farb-Nagpal in their paper

T. Church, J. Ellenberg, B. Farb, and R. Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984.

See Theorem C of that paper.

Another nice feature of these descriptions is that they generalize nicely to things beyond the symmetric groups. There is now a large literature on this, but a good starting point with many examples is my paper

A. Putman and S. Sam, Representation stability and finite linear groups, preprint 2014.

All of my papers can be downloaded from my webpage here and all of Church-Farb-etc's papers can be download from Church's webpage here.

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.