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 $p$.

arXiv comment59 pages

References in corpus (15)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• FI-modules over Noetherian rings
• GL-equivariant modules over polynomial rings in infinitely many variables
• Representation stability and finite linear groups
• Homological stability for configuration spaces of manifolds
• Introduction to twisted commutative algebras
• Homological stability for unordered configuration spaces
• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• Representations of categories of G-maps
• Representation Stability
• Homological stability for automorphism groups
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Improved homological stability for configuration spaces after inverting 2
• Uniformly Presented Vector Spaces

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• Periodicity in the cohomology of symmetric groups via divided powers
• Upper bounds of homological invariants of $F{I}_G$-modules
• Depth and the Local Cohomology of FI_G-modules
• Homological degrees of representations of categories with shift functors
• On the degree-wise coherence of FI_G-modules
• Filtrations and Homological degrees of FI-modules
• Bounding the homology of FI-modules
• Lattice Spectral Sequences and Cohomology of Configuration Spaces
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Two homological proofs of the Noetherianity of $F{I}_G$
• FI${}^m$-modules over Noetherian rings
• Stable representation theory: beyond the classical groups
• A Survey of Representation Stability Theory
• Generalized Representation Stability and FI_d-modules
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• On computing the eventual behavior of an FI-module over the rational numbers
• Sheaves of modules on atomic sites and discrete representations of topological groups
• Periodicity in the cohomology of finite general linear groups via q-divided powers
• Representation stability and arithmetic statistics of spaces of 0-cycles
• Asymptotic behaviors of representations of graded categories with inductive functors
• An inductive machinery for representations of categories with shift functors