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{\Large Department of Mathematics, BGU}

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{\Huge Colloquium}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, April 25, 2017}
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\textbf{At} \emph{14:30 -- 15:30}
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\textbf{In} \emph{Math -101}

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{\large\scshape Inna Entova-Aizenbud 
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  (BGU)
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will talk about
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{\Large\bfseries Stability in representation theory of the symmetric groups\par}
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\textsc{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:

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








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