Polynomial splitting measures and cohomology of the pure braid group

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.)

To appear in the Arnold Mathematical Journal

References in corpus (6)

• Representation Stability
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
• Splitting Behavior of $S_n$-Polynomials
• A family of measures on symmetric groups and the field with one element
• Stanley character polynomials

Cited by in corpus (3)

• Analytic number theory for 0-cycles
• Variations on the $S_n$-module $Li{e}_n$
• Factorization statistics and the twisted Grothendieck-Lefschetz formula