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Dan Petersen: Factorization statistics and bug-eyed configuration spaces

Time: Wed 2019-10-30 10.15

Location: Room 3418, Lindstedtsvägen 25. Department of Mathematics, KTH

Participating: Dan Petersen

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Fix a class function of the symmetric group Sn, 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 Sn-action on the cohomology of the configuration space of n points in R3 (!), but his argument gave no geometric reason for such a formula to exist. We give a geometric proof of Hyde's formula by studying the cohomology of a certain highly nonseparated scheme obtained by suitably gluing together complements of hyperplanes in the braid arrangement. (joint with Phil Tosteson)