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Report on the Broué–Malle–Rouquier Conjectures

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Perspectives in Lie Theory

Part of the book series: Springer INdAM Series ((SINDAMS,volume 19))

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Abstract

This paper is a short survey on the state-of-the-art concerning the main 1998 Broué–Malle–Rouquier conjectures about ‘Complex reflection groups, Braid groups, Hecke algebras’.

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References

  1. S. Ariki, On the semisimplicity of the Hecke algebra of \((\mathbb{Z}/r\mathbb{Z}) \wr \mathfrak{S}_{n}\). J. Algebra 169, 216–225 (1994)

    Article  MathSciNet  Google Scholar 

  2. S. Ariki, Representation theory of a Hecke algebra of G(r, p, n). J. Algebra 177, 164–185 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Ariki, K. Koike, A Hecke algebra of \((\mathbb{Z}/r\mathbb{Z}) \wr \mathfrak{S}_{n}\) and construction of its irreducible representations. Adv. Math. 106, 216–243 (1994)

    Article  MathSciNet  Google Scholar 

  4. E. Bannai, Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2. J. Math. Soc. Jpn. 28, 447–454 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Benard, Schur indices and splitting fields of the Unitary reflection groups. J. Algebra 38, 318–342 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Bessis, Zariski theorems and diagrams for braid groups. Invent. Math. 145, 487–507 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Bessis, Finite complex reflection arrangements are K(π, 1). Ann. Math. (2) 181, 809–904 (2015)

    Google Scholar 

  8. D. Bessis, J. Michel, Explicit presentations for exceptional braid groups. Exp. Math. 13, 257–266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Bigelow, Braid groups are linear. J. Am. Math. Soc. 14, 471–486 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Bremke, G. Malle, Reduced words and a length function for G(e, 1, n). Indag. Math. 8, 453–469 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Brieskorn, Die Fundamentalgruppe des Raumes der regulren Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57–61 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Broué, G. Malle, Zyklotomische Heckealgebren. Représentations unipotentes génériques et blocs des groupes réductifs finis. Astérisque 212, 119–189 (1993)

    MathSciNet  MATH  Google Scholar 

  13. M. Broué, G. Malle, R. Rouquier, On complex reflection groups and their associated braid groups, in CMS Conference Proceedings, vol. 16 (AMS, 1995)

    Google Scholar 

  14. M. Broué, G. Malle, R. Rouquier, Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)

    MathSciNet  MATH  Google Scholar 

  15. M. Broué, G. Malle, J. Michel, Towards spetses I. Transform. Groups 4, 157–218 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Callegaro, I. Marin, Homology computations for complex braid groups. J. Eur. Math. Soc. 16, 103–164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Chavli, The Broué-Malle-Rouquier conjecture for reflection groups of rank 2. Thèse de doctorat, Université Paris-Diderot (2016)

    Google Scholar 

  18. E. Chavli, Universal deformations of the finite quotients of the braid group on 3 strands. J. Algebra 459, 238–271 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Z. Chen, Flat connections and Brauer type algebras. J. Algebra 365, 114–146 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. CHEVIE, see http://webusers.imj-prg.fr/~jean.michel/chevie/index.html

  21. A. Cohen, D. Wales, Linearity of Artin groups of finite type. Isr. J. Math. 131, 101–123 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Dehornoy, Y. Lafont, Homology of Gaussian groups. Ann. Inst. Fourier (Grenoble) 53, 489–540 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Dehornoy, L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups. Proc. Lond. Math. Soc. 79(3), 569–604 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Dehornoy, F. Digne, E. Godelle, D. Krammer, J. Michel, Foundations of Garside theory, in EMS Tracts in Mathematics, vol. 22 (European Mathematical Society, Zurich, 2015)

    Book  MATH  Google Scholar 

  25. F. Digne, On the linearity of Artin braid groups. J. Algebra 268, 39–57 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. F. Digne, I. Marin, J. Michel, The center of pure complex braid groups. J. Algebra 347, 206–213 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Etingof, E. Rains, Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups. J. Algebra 299, 570–588 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Francis, Centralizers in the Hecke algebras of complex reflection groups. Preprint (2007), arxiv:0707.2822

    Google Scholar 

  29. L. Funar, On the quotients of cubic Hecke algebras. Commun. Math. Phys. 173, 513–558 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Krammer, Braid groups are linear. Ann. Math. (2) 155, 131–156 (2002)

    Google Scholar 

  31. G.I. Lehrer, D.E. Taylor, Unitary Reflection Groups (Cambridge University Press, Cambridge, 2009)

    MATH  Google Scholar 

  32. I. Losev, Finite-dimensional quotients of Hecke algebras. Algebra Number Theory 9, 493–502 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Malle, Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, in Finite Reductive Groups: Related Structures and Representations. Progress in Mathematics, vol. 141 (Birkhäuser, Basel, 1997), pp. 311–332

    Google Scholar 

  34. G. Malle, On the generic degrees of cyclotomic algebras. Represent. Theory 4, 342–369 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Malle, J. Michel, Constructing representations for Hecke algebras of complex reflection groups. LMS J. Comput. Math 13, 426–450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Marin, On the residual nilpotence of pure Artin groups. J. Group Theory 9, 483–485 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. I. Marin, Sur les représentations de Krammer génériques. Ann. Inst. Fourier (Grenoble) 57, 1883–1925 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. I. Marin, The cubic Hecke algebra on at most 5 strands. J. Pure Appl. Algebra 216, 2754–2782 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. I. Marin, Krammer representations for complex braid groups. J. Algebra 371, 175–206 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. I. Marin, The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G 26. J. Pure Appl. Algebra 218, 704–720 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. I. Marin, G. Pfeiffer, The BMR freeness conjecture for the 2-reflection groups. Math. Comput. 86, 2005–2023 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. I. Marin, E. Wagner, A cubic defining algebra for the Links-Gould polynomial. Adv. Math. 248, 1332–1365 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. I. Marin, E. Wagner, Markov traces on the BMW algebras. Preprint (2014), arXiv:1403.4021v1

    Google Scholar 

  44. J. Michel, The development version of the CHEVIE package of GAP3. J. Algebra 435, 308–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. T. Nakamura, A note on the K(π, 1) property of the orbit space of the unitary reflection group G(m, l, n). Sci. Papers College Arts Sci. Univ. Tokyo 33, 1–6 (1983)

    MathSciNet  MATH  Google Scholar 

  46. M. Picantin, Petits groupes gaussiens. Thèse de l’université de Caen, 2000

    Google Scholar 

  47. G.C. Shephard, J.A. Todd, Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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Marin, I. (2017). Report on the Broué–Malle–Rouquier Conjectures. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds) Perspectives in Lie Theory. Springer INdAM Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-58971-8_12

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