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Polynomial splitting measures and cohomology of the pure braid group. (English) Zbl 06855337
Summary: 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).

20F36 Braid groups; Artin groups
11R09 Polynomials (irreducibility, etc.)
11R32 Galois theory
11R34 Galois cohomology
20J06 Cohomology of groups
20G10 Cohomology theory for linear algebraic groups
12E20 Finite fields (field-theoretic aspects)
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
Full Text: DOI
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