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

Abstract

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. 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*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero 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 FIW-algebras finitely generated inFIW-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.