Stability in representation theory of the symmetric groups

In the finite-dimensional representation theory of the symmetric groups $S_n$ over the base field $\mathbb{C}$, there is an an interesting phenomena of “stabilization” as $n \to \infty$: some representations of $S_n$ appear in sequences $(V_n)_{n \geq 0}$, where each $V_n$ is a finite-dimensional representation of $S_n$, where $V_n$ become “the same” in a certain sense for $n >> 0$.

One manifestation of this phenomena are sequences $(V_n)_{n \geq 0}$ such that the characters of $S_n$ on $V_n$ are “polynomial in $n$”. More precisely, these sequences satisfy the condition: for $n>>0$, the trace (character) of the automorphism $\sigma \in S_n$ of $V_n$ is given by a polynomial in the variables $x_i$, where $x_i(\sigma)$ is the number of cycles of length $i$ in the permutation $\sigma$.

In particular, such sequences $(V_n)_{n \geq 0}$ satisfy the agreeable property that $\dim(V_n)$ is polynomial in $n$.

Such “polynomial sequences” are encountered in many contexts: cohomologies of configuration spaces of $n$ distinct ordered points on a connected oriented manifold, spaces of polynomials on rank varieties of $n \times n$ matrices, and more. These sequences are called $FI$-modules, and have been studied extensively by Church, Ellenberg, Farb and others, yielding many interesting results on polynomiality in $n$ of dimensions of these spaces.

A stronger version of the stability phenomena is described by the following two settings:

• The algebraic representations of the infinite symmetric group $$S_{\infty} = \bigcup_{n} S_n,$$ where each representation of $$S_{\infty}$$ corresponds to a ``polynomial sequence'' $$(V_n)_{n \geq 0}$$.
• The "polynomial" family of Deligne categories $$Rep(S_t), ~t \in \mathbb{C}$$, where the objects of the category $$Rep(S_t)$$ can be thought of as "continuations of sequences $$(V_n)_{n \geq 0}$$" to complex values of $$t=n$$.

I will describe both settings, show that they are connected, and explain some applications in the representation theory of the symmetric groups.