Factorization statistics and bug-eyed configuration spaces

Fix a class function of the symmetric group S_n, i.e. a function on the set of partitions of n. For any monic polynomial of degree n over a fixed finite field the degrees of its irreducible factors will form such a partition and we can ask about the average value of the class function evaluated on all such polynomials. Trevor Hyde ('18) proved by direct calculation that the answer can be expressed in terms of the S_n-action on the cohomology of the configuration space of n points in R^3 (!), but his argument gave no geometric reason for such a formula to exist. We give a geometric proof of Hyde's formula by applying the Grothendieck--Lefschetz trace formula to the cohomology of a certain highly nonseparated algebraic space obtained by gluing together complements of hyperplanes in the braid arrangement. (Joint with P. Tosteson)