On Domino Insertion and Kazhdan–Lusztig Cells in Type Bn

  • Cédric Bonnafé
  • Meinolf GeckEmail author
  • Lacrimioara Iancu
  • Thomas Lam
Part of the Progress in Mathematics book series (PM, volume 284)


Based on empirical evidence obtained using the CHEVIE computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan–Lusztig cells for type B n with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnafé and Iancu obtained earlier in the so-called asymptotic case. We give some partial results in support of our conjectures.


Coxeter groups Kazhdan-Lusztig cells Domino insertion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S.Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type G(m,1,n) and Kleshchev multipartitions, Osaka J. Math. 38 (2001), 827–837MathSciNetzbMATHGoogle Scholar
  2. 2.
    D.Barbasch and D.Vogan, Primitive ideals and orbital integrals on complex classical groups, Math. Ann. 259 (1982), 153–199MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    C.Bonnafé, Two-sided cells in type B in the asymptotic case, J. Algebra 304 (2006), 216–236MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    C.Bonnafé and C.Hohlweg, Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier 56 (2006), 131–181zbMATHCrossRefGoogle Scholar
  5. 5.
    C.Bonnafé and L.Iancu, Left cells in type B n with unequal parameters, Represent. Theor. 7 (2003), 587–609zbMATHCrossRefGoogle Scholar
  6. 6.
    R.W.Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, NewYork,1985zbMATHGoogle Scholar
  7. 7.
    F.du Cloux, The Coxeter programme, version3.0; electronically available at
  8. 8.
    R.Dipper, G. D.James and G.E.Murphy, Hecke algebras of type B n at roots of unity, Proc. London Math. Soc. 70 (1995), 505–528MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R.Dipper, G.James and G.E.Murphy, Gram determinants of type B n, J. Algebra 189 (1997), 481–505MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D.Garfinkle, On the classification of primitive ideals for complex classical Lie algebras I, Compositio Math. 75 (1990), 135–169MathSciNetzbMATHGoogle Scholar
  11. 11.
    D.Garfinkle, On the classification of primitive ideals for complex classical Lie algebras II, Compositio Math. 81 (1992), 307–336MathSciNetzbMATHGoogle Scholar
  12. 12.
    M.Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theor. 6 (2002), 1–30 (electronic)Google Scholar
  13. 13.
    M.Geck, On the induction of Kazhdan–Lusztig cells, Bull. London Math. Soc. 35 (2003), 608–614MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M.Geck, Computing Kazhdan–Lusztig cells for unequal parameters, J. Algebra 281 (2004), 342–365MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    M.Geck, Left cells and constructible representations, Represent. Theor. 9 (2005), 385–416 (electronic)Google Scholar
  16. 16.
    M.Geck, Relative Kazhdan–Lusztig cells, Represent. Theor. 10 (2006), 481–524MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    M.Geck, Modular principal series representations, Int. Math. Res. Notices 2006, 1–20, Article ID 41957Google Scholar
  18. 18.
    M.Geck and L.Iancu, Lusztig’s a-function in type B n in the asymptotic case. Special issue celebrating the 60th birthday of George Lusztig, Nagoya J. Math. 182 (2006), 199–240Google Scholar
  19. 19.
    M.Geck and N.Jacon, Canonical basic sets in type B. Special issue celebrating the 60th birthday of Gordon James, J. Algebra 306 (2006), 104–127Google Scholar
  20. 20.
    M.Geck and G.Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Math. Soc. Monographs, New Series 21, pp. xvi+446. Oxford University Press, NewYork, 2000Google Scholar
  21. 21.
    M.Geck, G.Hiss, F.Lübeck, G.Malle, and G.Pfeiffer, CHEVIE – A system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210; electronically available at
  22. 22.
    M.Geck, L.Iancu and C.Pallikaros, Specht modules and Kazhdan–Lusztig cells in type B n, J. Pure Appl. Algebra 212 (2008), 1310–1320MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    I.G.Gordon and M.Martino, Calogero–Moser space, reduced rational Cherednik algebras and two-sided cells, preprint (2007), available at
  24. 24.
    J.J.Graham and G.I.Lehrer, Cellular algebras, Invent.Math. 123 (1996), 1–34MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M.D.Haiman, On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theor. Ser. A 50 (1989), 196–225MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    D.A.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    T.Lam, Growth diagrams, domino insertion, and sign-imbalance, J. Comb. Theor. Ser. A., 107 (2004), 87–115zbMATHCrossRefGoogle Scholar
  28. 28.
    M.vanLeeuwen, The Robinson–Schensted and Schutzenberger algorithms, an elementary approach, The Foata Festschrift, Electron. J. Combin. 3 (1996), Research Paper 15Google Scholar
  29. 29.
    G.Lusztig, Left cells in Weyl groups, Lie Group Representations, I (eds R.L. R.Herb and J.Rosenberg), Lecture Notes in Mathematics 1024, pp.99–111. Springer, Berlin, 1983Google Scholar
  30. 30.
    G.Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser.18. AMS, Providence, RI, 2003Google Scholar
  31. 31.
    T.Pietraho, Equivalence classes in the Weyl groups of type B n, J. Algebraic Combinatorics 27 (2008), 247–262MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    T.Pietraho, A relation for domino Robinson–Schensted Algorithms, Annals of Combinatorics (to appear)Google Scholar
  33. 33.
    M.Shimozono and D.E.White, Color-to-spin ribbon Schensted algorithms, Discrete Math. 246 (2002), 295–316MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    R.Stanley, Enumerative Combinatorics, vol 2. Cambridge Univerity Press, London, 1999CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Cédric Bonnafé
  • Meinolf Geck
    • 1
    Email author
  • Lacrimioara Iancu
  • Thomas Lam
  1. 1.Institute of Mathematics, King’s CollegeAberdeen UniversityAberdeenUK

Personalised recommendations