# Stability in representation theory of the symmetric groups

Seminar
Speaker
Dr. Inna Entova-Aizenbud (Ben-Gurion University)
Date
03/05/2017 - 12:00 - 11:00Add to Calendar 2017-05-03 11:00:00 2017-05-03 12:00:00 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. Third floor seminar room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room
Abstract

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 Updated Date : 02/05/2017