Factorization statistics and bugeyed 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 GrothendieckLefschetz 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 nonHausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.06024
 arXiv:
 arXiv:2004.06024
 Bibcode:
 2020arXiv200406024P
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Number Theory;
 Mathematics  Representation Theory
 EPrint:
 19 pages. v2: added reference to prior work of Proudfoot v3: final version to appear in G&