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- W2607140410 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." @default.
- W2607140410 created "2017-04-28" @default.
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- W2607140410 date "2017-03-18" @default.
- W2607140410 modified "2023-10-06" @default.
- W2607140410 title "Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields" @default.
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