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.)

arXiv commentTo appear in the Arnold Mathematical Journal

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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