Advertisement

Arnold Mathematical Journal

, Volume 3, Issue 2, pp 219–249 | Cite as

Polynomial Splitting Measures and Cohomology of the Pure Braid Group

  • Trevor HydeEmail author
  • Jeffrey C. Lagarias
Research Contribution

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).

Keywords

Symmetric groups Braid group Configuration space 

Mathematics Subject Classification

Primary 11R09 Secondary 11R32 12E20 12E25 

Notes

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.

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)]Google Scholar
  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)Google Scholar
  3. Callegaro, F., Gaiffi, G.: On models of the braid arrangement and their hidden symmetries. Int. Math. Res. Not. 21, 11117–11149 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen, W.: Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting (2016). eprint: arXiv:1603.03931
  5. Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  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)Google Scholar
  7. Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  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)Google Scholar
  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)Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Farb, B.: Representation stability. In: Proceedings of the 2014 ICM, Seoul, Korea. eprint: arXiv:1404.4065
  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)MathSciNetCrossRefzbMATHGoogle Scholar
  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)Google Scholar
  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)Google Scholar
  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
  16. Kisin, M., Lehrer, G.I.: Equivariant Poincaré polynomials and counting points over finite fields. J. Algebra 247(2), 435–451 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lagarias, J.C.: A family of measures on symmetric groups and the field with one element. J. Number Theory 161, 311–342 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lagarias, J.C., Weiss, B.L.: Splitting behavior of \(S_n\) polynomials. Res. Number Theory 1, paper 9, 30 pp. (2015)Google Scholar
  19. Lehrer, G.I.: On the Poincaré series associated with Coxeter group actions on complements of hyperplanes. J. Lond. Math. Soc. 36(2), 275–294 (1987)CrossRefzbMATHGoogle Scholar
  20. Lehrer, G.I.: The \(\ell \)-adic cohomology of hyperplane complements. Bull. Lond. Math. Soc. 24, 76–82 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Lehrer, G.I., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104(2), 410–424 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  23. Mathieu, O.: Hidden \(\Sigma _{n+1}\)-actions. Commun. Math. Phys. 176, 467–474 (1996)MathSciNetCrossRefGoogle Scholar
  24. Metropolis, N., Rota, G.-C.: Witt vectors and the algebra of necklaces. Adv. Math. 50, 95–125 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  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)Google Scholar
  26. Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56, 57–89 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der math. Wiss, vol. 300. Springer, Berlin (1992)Google Scholar
  28. Rosen, M.: Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210. Springer, New York (2002)Google Scholar
  29. Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, vol. 203. Springer Science and Business Media, Berlin (2013)zbMATHGoogle Scholar
  30. Śniady, P.: Stanley character polynomials. In: The Mathematical Legacy of Richard P. Stanley, vol. 100, p. 323 (2016)Google Scholar
  31. Stanley, R.P.: Some aspects of groups acting on finite posets. J. Comb. Theory Ser. A 32, 132–161 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  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]Google Scholar
  33. Sundaram, S.: The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice. Adv. Math. 104, 225–296 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Sundaram, S., Welker, V.: Group actions on arrangements of linear subspaces and applications to configuration spaces. Trans. Am. Math. Soc. 349(4), 1389–1420 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Weiss, B.L.: Probabilistic Galois theory over \(p\)-adic fields. J. Number Theory 133, 1537–1563 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Yuzvinsky, S.: Orlik-Solomon algebras in algebra, topology and geometry. Russ. Math. Surv. 56, 294–364 (2001)CrossRefzbMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations