  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.)
