Factorization statistics and bug-eyed configuration spaces

Abstract

A recent theorem of Hyde proves that the factorizations statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in ${\mathbb{R}}^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck--Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to an arbitrary Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

arXiv comment19 pages. v2: added reference to prior work of Proudfoot v3: final version to appear in G&T

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Cited by in corpus (2)

• Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
• Eulerian representations for real reflection groups