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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).

##### MSC:
 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
##### Keywords:
symmetric groups; braid group; configuration space
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