Polynomial splitting measures and cohomology of the pure braid group

TL;DR

This paper connects polynomial splitting measures over finite fields to the cohomology of pure braid groups, revealing that these measures are related to characters of symmetric group representations.

Contribution

It expresses splitting measure coefficients in terms of cohomology characters, linking algebraic combinatorics with topological group cohomology.

Findings

Splitting measures are Laurent polynomials with coefficients linked to cohomology characters.

For specific parameter values, measures correspond to actual or virtual symmetric group representations.

The work bridges polynomial factorization probabilities with the cohomology of pure braid groups.

Abstract

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)