[Paper Review] Representation stability, secondary stability, and polynomial functors

Jeremy Miller, Peter Patzt+1 more|arXiv (Cornell University)|Oct 12, 2019

Homotopy and Cohomology in Algebraic Topology40 references6 citations

TL;DR

This paper establishes representation stability and secondary homological stability for families of groups with polynomial coefficient systems by introducing a framework based on polynomial functors and derived representation stability. It proves that polynomial coefficients imply derived representation stability, enabling new stability theorems for hyperelliptic mapping class groups, congruence subgroups, diffeomorphism groups, and general linear groups of the sphere spectrum with improved stable ranges.

ABSTRACT

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum.

Motivation & Objective

• To develop a general framework for representation stability and secondary homological stability with polynomial coefficient systems.
• To prove that polynomial coefficient systems imply derived representation stability, enabling stability theorems beyond classical settings.
• To extend classical homological stability theorems to twisted coefficients using polynomiality as a sufficient condition.
• To apply the framework to new families of groups, including hyperelliptic mapping class groups and diffeomorphism groups of surfaces.
• To improve the stable range for homological stability of GL_n(𝕊), the general linear group of the sphere spectrum.

Proposed method

• Uses polynomial functors and polynomial coefficient systems to model twisted coefficients in homological stability.
• Applies central stability complexes and Koszul resolutions to analyze derived representation stability.
• Introduces a spectral sequence argument with triple complexes to relate homological stability to representation stability.
• Replaces derived representation stability assumptions with polynomiality via cohomological techniques and Tor computations.
• Leverages the fact that polynomiality implies derived representation stability to simplify spectral sequences.
• Uses the cone construction in equivariant homotopy theory to measure stability, adapted to representation-theoretic settings.

Experimental results

Research questions

• RQ1 Can polynomial coefficient systems ensure derived representation stability in families of groups with group actions?
• RQ2 Does polynomiality of coefficients suffice to establish representation stability and secondary homological stability for twisted homology?
• RQ3 How can spectral sequences with polynomial coefficients be simplified to prove stability theorems?
• RQ4 Can the framework be applied to hyperelliptic mapping class groups and congruence subgroups with twisted coefficients?
• RQ5 What improvements in stable range can be achieved for homological stability of GL_n(𝕊) using polynomial functors?

Key findings

• Polynomial coefficient systems imply derived representation stability, providing a sufficient condition for representation stability in twisted settings.
• The framework proves representation stability for congruence subgroups with polynomial coefficients, extending known results.
• Secondary homological stability is established for diffeomorphism groups of surfaces viewed as discrete groups.
• Homological stability for GL_n(𝕊) is improved with a better stable range using polynomial functors.
• The spectral sequence argument converges to zero under polynomiality, enabling new stability theorems via simplified E^1 pages.
• The method generalizes classical homological stability theorems to include twisted coefficients via polynomiality, unifying several stability phenomena.