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We study for each n a one-parameter family of complex-valued measures on the symmetric group $$S_n$$ S n , which interpolate the probability of a monic, degree n, square-free polynomial in $$\mathbb {F}_q[x]$$ F q [ x ] having a given factorization type. For a fixed factorization type, indexed by a partition $$\lambda $$ λ of n, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $$S_n$$ S n -subrepresentations of the cohomology of the pure braid group $$H^{\bullet }(P_n, \mathbb {Q})$$ H ∙ ( P n , Q ) . We deduce that the splitting measures for all parameter values $$z= -\frac{1}{m}$$ z = - 1 m (resp. $$z= \frac{1}{m}$$ z = 1 m ), after rescaling, are characters of $$S_n$$ S n -representations (resp. virtual $$S_n$$ S n -representations).
Arnold Mathematical Journal – Springer Journals
Published: Mar 6, 2017
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