FI-modules and the cohomology of modular representations of symmetric groups

Abstract

An FI-module $V$ over a commutative ring $\mathbf{k}$ encodes a sequence $(V_n{)}_{n\ge 0}$ of representations of the symmetric groups $({\mathfrak{S}}_n{)}_{n\ge 0}$ over $\mathbf{k}$. 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 $\ge 2$ and ${conf}_n(\mathcal{M})$ is the configuration space of unordered $n$-tuples of distinct points in $\mathcal{M}$ then the mod-$p$ cohomology groups $H^t({conf}_n(\mathcal{M}),\mathbf{k})$ are eventually periodic in $n$ with period a power of $p$.

59 pages

References in corpus (3)

• Introduction to twisted commutative algebras
• Representation Stability
• Uniformly Presented Vector Spaces

Cited by in corpus (8)

• The geometry of polynomial representations
• Bounding the homology of FI-modules
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• FI${}^m$-modules over Noetherian rings
• Stable representation theory: beyond the classical groups
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• Representation stability and arithmetic statistics of spaces of 0-cycles
• Sheaves of modules on atomic sites and discrete representations of topological groups