# 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.

- Last modified: 2/05/2017