FI-modules and the cohomology of modular representations of symmetric groups
By Rohit Nagpal. 2015
48 citations recorded by the source corpus.
An FI-module $V$ over a commutative ring $\bf{k}$ encodes a sequence $(V_n)_{n \geq 0}$ of representations of the symmetric groups $(\mathfrak{S}_n)_{n \geq 0}$ over $\bf{k}$. In this paper, we show that for a "finitely generated" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\geq 2$ and $\mathit{conf}_n(\mathcal{M})$ is the configuration space of unordered $n$-tuples of distinct points in $\mathcal{M}$ then the mod-$p$ cohomology groups $H^{t}(\mathit{conf}_n(\mathcal{M}),\bf{k})$ are eventually periodic in $n$ with period a power of $p$.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9d7c51495cbe996f8a9de36307a93b5bb35dcd0c
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Correction to the article A spectral sequence for stratified spaces and configuration spaces of points (2016, result): “-module over Z gives a homological stability result with rational coecients, but it also has interesting consequences for the mod phomology of the unordered conguration spaces: by results of Nagpal [Nag15], our theorem implies that the groups HBM i+2dn(F A(M;n)=Sn;Fp) become eventually periodic. Vak”