# Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

## TL;DR

This paper connects the topology of type B/C hyperplane complements with polynomial statistics over finite fields, proving stability results and extending known type A theorems using algebraic and combinatorial methods.

## Contribution

It establishes a type B/C analogue of a stability theorem for polynomial statistics, introduces a new proof of the type A result, and analyzes the algebraic structure of cohomology algebras.

## Key findings

- Proves a stability result for polynomial statistics over finite fields.
- Establishes a correspondence between cohomology and polynomial statistics.
- Shows that FI_W-algebra structure alone is insufficient for convergence proofs.

## Abstract

In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over $\mathbb{F}_q$ with nonzero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FI$_\mathcal{W}$-algebras finitely generated in FI$_\mathcal{W}$- degree $2$, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.