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).
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Acknowledgements
We thank Richard Stanley for raising a question about the relation of the braid group cohomology to the regular representation, answered by Theorem 1.4. We thank Weiyan Chen for pointing out to us that Theorem 1.1 is shown in Lehrer (1987) and for subsequently bringing the work of Gaiffi (1996) to our attention. We thank Philip Tosteson and John Wiltshire-Gordon for helpful conversations. We thank the reviewers for helpful comments.
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Work of J. C. Lagarias was partially supported by NSF Grant DMS-1401224.
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Hyde, T., Lagarias, J.C. Polynomial Splitting Measures and Cohomology of the Pure Braid Group. Arnold Math J. 3, 219–249 (2017). https://doi.org/10.1007/s40598-017-0064-z
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DOI: https://doi.org/10.1007/s40598-017-0064-z