×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnol’d, V.I.: The cohomology ring of the colored braid group. Math. Notes 5, 138-140 (1969) [English translation of: Mat. Zametki 5, 227-231 (1969)] · Zbl 1339.55004
[2] Bhargava, M.: Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants. Int. Math. Res. Not. 17, Art. ID rnm052, 20 pp. (2007) · Zbl 1145.11080
[3] Callegaro, F; Gaiffi, G, On models of the braid arrangement and their hidden symmetries, Int. Math. Res. Not., 21, 11117-11149, (2015) · Zbl 1349.14171
[4] Chen, W.: Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting (2016). eprint: arXiv:1603.03931 · Zbl 1033.52019
[5] Church, T; Farb, B, Representation theory and homological stability, Adv. Math., 245, 250-314, (2013) · Zbl 1300.20051
[6] Church, T., Ellenberg, J.S., Farb, B.: Representation stability in cohomology and asymptotics for families of varieties over finite fields. In: Algebraic Topology: Applications and New Directions, pp. 1-54. Contemporary Mathematics, vol. 620. American Mathematical Society, Providence (2014) · Zbl 1388.14148
[7] Church, T; Ellenberg, JS; Farb, B, FI-modules and stability for representations of symmetric groups, Duke Math. J., 164, 1833-1910, (2015) · Zbl 1339.55004
[8] Dedekind, R.: Über Zusammenhang zwischen der Theorie der Ideale und der Theorie der höhere Kongruenzen, Abh. König. Ges. der Wissen. zu Göttingen 23, 1-23 (1878) · Zbl 0496.06001
[9] Dimca, A., Yuzvinsky, S.: Lectures on Orlik-Solomon algebras. In: Arrangements, Local Systems and Singularities. Progress in Mathematics, vol. 283, pp. 83-110. Birkhäuser, Basel (2010) · Zbl 1285.52012
[10] Dołega, M; Féray, V; Śniady, P, Explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations, Adv. Math., 225, 81-120, (2010) · Zbl 1231.05283
[11] Farb, B.: Representation stability. In: Proceedings of the 2014 ICM, Seoul, Korea. eprint: arXiv:1404.4065 · Zbl 1373.55006
[12] Gaiffi, G, The actions of \(S_{n+1}\) and \(S_n\) on the cohomology ring of a Coxeter arrangement of type \(A_{n-1}\), Manuscr. math., 91, 83-94, (1996) · Zbl 0886.57029
[13] Getzler, E.: Operads and moduli spaces of genus \(0\) Riemann surfaces. In: The Moduli Space of Curves (Texel Island 1994), pp. 199-230. Progress in Mathematics, vol. 129. Birkhäuser, Boston (1995) · Zbl 0858.58016
[14] Grothendieck, A.: Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à11, Volume 1960/61 of Séminaire de Géomeétrie Albebrique. IHES, Paris (1963)
[15] Hersh, P., Reiner, V.: Representation stability for cohomology of configuration spaces in \({\mathbb{R}}^d\) (Appendix joint with Steven Sam). In: International Mathematics Research Notices (2015). doi:10.1093/imrn/rnw060. eprint: arXiv:1505.04196v3 · Zbl 0649.20041
[16] Kisin, M; Lehrer, GI, Equivariant Poincaré polynomials and counting points over finite fields, J. Algebra, 247, 435-451, (2002) · Zbl 1039.14005
[17] Lagarias, JC, A family of measures on symmetric groups and the field with one element, J. Number Theory, 161, 311-342, (2016) · Zbl 1337.11074
[18] Lagarias, J.C., Weiss, B.L.: Splitting behavior of \(S_n\) polynomials. Res. Number Theory 1, paper 9, 30 pp. (2015) · Zbl 0545.05009
[19] Lehrer, GI, On the Poincaré series associated with Coxeter group actions on complements of hyperplanes, J. Lond. Math. Soc., 36, 275-294, (1987) · Zbl 0649.20041
[20] Lehrer, GI, The \(ℓ \)-adic cohomology of hyperplane complements, Bull. Lond. Math. Soc., 24, 76-82, (1992) · Zbl 0770.14013
[21] Lehrer, GI; Solomon, L, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra, 104, 410-424, (1986) · Zbl 0608.20010
[22] Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995) · Zbl 0824.05059
[23] Mathieu, O, Hidden \(Σ _{n+1}\)-actions, Commun. Math. Phys., 176, 467-474, (1996) · Zbl 0858.58016
[24] Metropolis, N; Rota, G-C, Witt vectors and the algebra of necklaces, Adv. Math., 50, 95-125, (1983) · Zbl 0545.05009
[25] Moreau, C.: Sur les permutations circulaires distinctes. Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2(11), 309-314 (1872)
[26] Orlik, P; Solomon, L, Combinatorics and topology of complements of hyperplanes, Invent. Math., 56, 57-89, (1980) · Zbl 0432.14016
[27] Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der math. Wiss, vol. 300. Springer, Berlin (1992) · Zbl 0757.55001
[28] Rosen, M.: Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210. Springer, New York (2002) · Zbl 1043.11079
[29] Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, vol. 203. Springer Science and Business Media, Berlin (2013) · Zbl 0823.05061
[30] Śniady, P.: Stanley character polynomials. In: The Mathematical Legacy of Richard P. Stanley, vol. 100, p. 323 (2016) · Zbl 1368.20007
[31] Stanley, RP, Some aspects of groups acting on finite posets, J. Comb. Theory Ser. A, 32, 132-161, (1982) · Zbl 0496.06001
[32] Stanley, R.P.: Enumerative combinatorics, vol. 1. In: Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997) [Corrected reprint of the 1986 original] · Zbl 0608.05001
[33] Sundaram, S, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math., 104, 225-296, (1994) · Zbl 0823.05063
[34] Sundaram, S; Welker, V, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Am. Math. Soc., 349, 1389-1420, (1997) · Zbl 0945.05067
[35] Weiss, BL, Probabilistic Galois theory over \(p\)-adic fields, J. Number Theory, 133, 1537-1563, (2013) · Zbl 1300.11120
[36] Yuzvinsky, S, Orlik-Solomon algebras in algebra, topology and geometry, Russ. Math. Surv., 56, 294-364, (2001) · Zbl 1033.52019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.