Quasi-isolated blocks and Brauer’s height zero conjecture

Abstract

This paper has two main results. Firstly, we complete the parametrisation of all $p$-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer’s long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups.

  • [AB79] Go to document J. Alperin and M. Broué, "Local methods in block theory," Ann. of Math., vol. 110, iss. 1, pp. 143-157, 1979.
    @article {AB79, MRKEY = {0541333},
      AUTHOR = {Alperin, J. and Brou{é},
      Michel},
      TITLE = {Local methods in block theory},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {110},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {143--157},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20C99},
      MRNUMBER = {0541333},
      MRREVIEWER = {Wolfgang Willems},
      DOI = {10.2307/1971248},
      ZBLNUMBER = {0416.20006},
      }
  • [AKO] Go to document M. Aschbacher, R. Kessar, and B. Oliver, Fusion Systems in Algebra and Topology, Cambridge: Cambridge Univ. Press, 2011, vol. 391.
    @book {AKO, MRKEY = {2848834},
      AUTHOR = {Aschbacher, Michael and Kessar, Radha and Oliver, Bob},
      TITLE = {Fusion Systems in Algebra and Topology},
      SERIES = {London Math. Soc. Lect. Note Series},
      VOLUME = {391},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2011},
      PAGES = {vi+320},
      ISBN = {978-1-107-60100-0},
      MRCLASS = {20C20 (20D20 55P99)},
      MRNUMBER = {2848834},
      MRREVIEWER = {Charles W. Eaton},
      DOI = {10.1017/CBO9781139003841},
      ZBLNUMBER = {1255.20001},
      }
  • [BK] Go to document T. R. Berger and R. Knörr, "On Brauer’s height $0$ conjecture," Nagoya Math. J., vol. 109, pp. 109-116, 1988.
    @article {BK, MRKEY = {0931954},
      AUTHOR = {Berger, T. R. and Kn{ö}rr, R.},
      TITLE = {On {B}rauer's height {$0$} conjecture},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {109},
      YEAR = {1988},
      PAGES = {109--116},
      ISSN = {0027-7630},
      CODEN = {NGMJA2},
      MRCLASS = {20C20},
      MRNUMBER = {0931954},
      MRREVIEWER = {David Gluck},
      URL = {http://projecteuclid.org/euclid.nmj/1118780894},
      ZBLNUMBER = {0637.20006},
      }
  • [BE] Go to document H. I. Blau and H. Ellers, "Brauer’s height zero conjecture for central quotients of special linear and special unitary groups," J. Algebra, vol. 212, iss. 2, pp. 591-612, 1999.
    @article {BE, MRKEY = {1676855},
      AUTHOR = {Blau, Harvey I. and Ellers, Harald},
      TITLE = {Brauer's height zero conjecture for central quotients of special linear and special unitary groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {212},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {591--612},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G05 (20C20 20G40)},
      MRNUMBER = {1676855},
      MRREVIEWER = {Julianne G. Rainbolt},
      DOI = {10.1006/jabr.1998.7661},
      ZBLNUMBER = {0930.20008},
      }
  • [B05] Go to document C. Bonnafé, "Quasi-isolated elements in reductive groups," Comm. Algebra, vol. 33, iss. 7, pp. 2315-2337, 2005.
    @article {B05, MRKEY = {2153225},
      AUTHOR = {Bonnaf{é},
      C{é}dric},
      TITLE = {Quasi-isolated elements in reductive groups},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {33},
      YEAR = {2005},
      NUMBER = {7},
      PAGES = {2315--2337},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {20G15},
      MRNUMBER = {2153225},
      MRREVIEWER = {Evgenii L. Bashkirov},
      DOI = {10.1081/AGB-200063602},
      ZBLNUMBER = {1096.20037},
      }
  • [B06] C. Bonnafé, Sur les Caractères des Groupes Réductifs Finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires, Paris: Soc. Math. France, 2006, vol. 306.
    @book {B06, MRKEY = {2274998},
      AUTHOR = {Bonnaf{é},
      C{é}dric},
      TITLE = {Sur les Caractères des Groupes Réductifs Finis à Centre Non Connexe: Applications aux Groupes Spéciaux Linéaires et Unitaires},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {306},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      YEAR = {2006},
      PAGES = {vi+165},
      ISSN = {0303-1179},
      MRCLASS = {20G40 (20G05)},
      MRNUMBER = {2274998},
      MRREVIEWER = {Robert B{é}dard},
      ZBLNUMBER = {1157.20022},
      }
  • [BMi] Go to document C. Bonnafé and J. Michel, "Computational proof of the Mackey formula for $q>2$," J. Algebra, vol. 327, pp. 506-526, 2011.
    @article {BMi, MRKEY = {2746047},
      AUTHOR = {Bonnaf{é},
      C{é}dric and Michel, Jean},
      TITLE = {Computational proof of the {M}ackey formula for {$q>2$}},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {327},
      YEAR = {2011},
      PAGES = {506--526},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20C33 (20G05 20G40)},
      MRNUMBER = {2746047},
      MRREVIEWER = {Simon M. Goodwin},
      DOI = {10.1016/j.jalgebra.2010.10.030},
      ZBLNUMBER = {1231.20041},
      }
  • [BR] Go to document C. Bonnafé and R. Rouquier, "Catégories dérivées et variétés de Deligne-Lusztig," Publ. Math. Inst. Hautes Études Sci., vol. 97, pp. 1-59, 2003.
    @article {BR, MRKEY = {2010739},
      AUTHOR = {Bonnaf{é},
      C{é}dric and Rouquier, Rapha{ë}l},
      TITLE = {Catégories dérivées et variétés de {D}eligne-{L}usztig},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      VOLUME = {97},
      YEAR = {2003},
      PAGES = {1--59},
      ISSN = {0073-8301},
      MRCLASS = {20G05 (18E30 20G40)},
      MRNUMBER = {2010739},
      MRREVIEWER = {Jean Michel},
      DOI = {10.1007/s10240-003-0013-3},
      ZBLNUMBER = {1054.20024},
      }
  • [Br41] Go to document R. Brauer, "Investigations on group characters," Ann. of Math., vol. 42, pp. 936-958, 1941.
    @article {Br41, MRKEY = {0005731},
      AUTHOR = {Brauer, Richard},
      TITLE = {Investigations on group characters},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {42},
      YEAR = {1941},
      PAGES = {936--958},
      ISSN = {0003-486X},
      MRCLASS = {20.0X},
      MRNUMBER = {0005731},
      MRREVIEWER = {G. de B. Robinson},
      ZBLNUMBER = {0061.03702},
      DOI = {10.2307/1968775},
     }
  • [Br55] R. Brauer, "Number theoretical investigations on groups of finite order," in Proceedings of the International Symposium on Algebraic Number Theory, Tokyo, 1956, pp. 55-62.
    @inproceedings {Br55, MRKEY = {0083490},
      AUTHOR = {Brauer, Richard},
      TITLE = {Number theoretical investigations on groups of finite order},
      BOOKTITLE = {Proceedings of the International Symposium on Algebraic Number Theory},
      VENUE={{T}okyo and {N}ikko, 1955},
      PAGES = {55--62},
      PUBLISHER = {Science Council of Japan},
      ADDRESS = {Tokyo},
      YEAR = {1956},
      MRCLASS = {20.0X},
      MRNUMBER = {0083490},
      MRREVIEWER = {G. de B. Robinson},
      ZBLNUMBER = {0073.01403},
      }
  • [Br71] Go to document R. Brauer, "Some applications of the theory of blocks of characters of finite groups. IV," J. Algebra, vol. 17, pp. 489-521, 1971.
    @article {Br71, MRKEY = {0281806},
      AUTHOR = {Brauer, Richard},
      TITLE = {Some applications of the theory of blocks of characters of finite groups. {IV}},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {17},
      YEAR = {1971},
      PAGES = {489--521},
      ISSN = {0021-8693},
      MRCLASS = {20.80},
      MRNUMBER = {0281806},
      MRREVIEWER = {W. Feit},
      ZBLNUMBER = {0247.20013},
      DOI = {10.1016/0021-8693(71)90006-8},
     }
  • [BMM] M. Broué, G. Malle, and J. Michel, "Generic blocks of finite reductive groups," in Représentations Unipotentes Génériques et Blocs des Groupes Réductifs Finis, Paris: Soc. Math. France, 1993, vol. 212, pp. 7-92.
    @incollection {BMM, MRKEY = {1235832},
      AUTHOR = {Brou{é},
      Michel and Malle, Gunter and Michel, Jean},
      TITLE = {Generic blocks of finite reductive groups},
      BOOKTITLE = {Repr{é}sentations Unipotentes G{é}n{é}riques et Blocs des Groupes R{é}ductifs Finis},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {212},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      YEAR = {1993},
      PAGES = {7--92},
      ISSN = {0303-1179},
      MRCLASS = {20G05 (20C20)},
      MRNUMBER = {1235832},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0843.20012},
      }
  • [BM93] M. Broué and J. Michel, "Blocs à groupes de défaut abéliens des groupes réductifs finis," in Représentations Unipotentes Génériques et Blocs des Groupes Réductifs Finis, Paris: Soc. Math. France, 1993, vol. 212, pp. 93-117.
    @incollection {BM93, MRKEY = {1235833},
      AUTHOR = {Brou{é},
      Michel and Michel, Jean},
      TITLE = {Blocs à groupes de défaut abéliens des groupes réductifs finis},
      BOOKTITLE = {Repr{é}sentations Unipotentes G{é}n{é}riques et Blocs des Groupes R{é}ductifs Finis},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      VOLUME = {212},
      YEAR = {1993},
      PAGES = {93--117},
      ISSN = {0303-1179},
      MRCLASS = {20C20 (20D06 20G40)},
      MRNUMBER = {1235833},
      MRREVIEWER = {R. W. Carter},
      ZBLNUMBER = {0832.20024},
      }
  • [CE93] Go to document M. Cabanes and M. Enguehard, On general blocks of finite reductive groups: ordinary characters and defect groups, 1993.
    @misc{CE93,
      author = {Cabanes, Marc and Enguehard, Michel},
      TITLE = {On general blocks of finite reductive groups: ordinary characters and defect groups},
      NOTE={Rapport de Recherche du LMENS 93-13},
      YEAR={1993},
      URL = {http://www.math.jussieu.fr/~cabanes/CabEngLMENS1993.pdf},
     }
  • [CE94] Go to document M. Cabanes and M. Enguehard, "On unipotent blocks and their ordinary characters," Invent. Math., vol. 117, iss. 1, pp. 149-164, 1994.
    @article {CE94, MRKEY = {1269428},
      AUTHOR = {Cabanes, Marc and Enguehard, Michel},
      TITLE = {On unipotent blocks and their ordinary characters},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {117},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {149--164},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20G05 (20C20)},
      MRNUMBER = {1269428},
      MRREVIEWER = {Bhama Srinivasan},
      DOI = {10.1007/BF01232237},
      ZBLNUMBER = {0817.20046},
      }
  • [CE99] Go to document M. Cabanes and M. Enguehard, "On blocks of finite reductive groups and twisted induction," Adv. Math., vol. 145, iss. 2, pp. 189-229, 1999.
    @article {CE99, MRKEY = {1704575},
      AUTHOR = {Cabanes, Marc and Enguehard, Michel},
      TITLE = {On blocks of finite reductive groups and twisted induction},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {145},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {189--229},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {20C33 (20C20 20G40)},
      MRNUMBER = {1704575},
      MRREVIEWER = {Bhama Srinivasan},
      DOI = {10.1006/aima.1998.1814},
      ZBLNUMBER = {0954.20023},
      }
  • [CE] Go to document M. Cabanes and M. Enguehard, Representation Theory of Finite Reductive Groups, Cambridge: Cambridge Univ. Press, 2004, vol. 1.
    @book {CE, MRKEY = {2057756},
      AUTHOR = {Cabanes, Marc and Enguehard, Michel},
      TITLE = {Representation Theory of Finite Reductive Groups},
      SERIES = {New Math. Monogr.},
      VOLUME = {1},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      PAGES = {xviii+436},
      ISBN = {0-521-82517-2},
      MRCLASS = {20G05 (20G40)},
      MRNUMBER = {2057756},
      MRREVIEWER = {Bhama Srinivasan},
      DOI = {10.1017/CBO9780511542763},
      ZBLNUMBER = {1069.20032},
      }
  • [Atl] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Eynsham: Oxford University Press, 1985.
    @book {Atl, MRKEY = {0827219},
      AUTHOR = {Conway, J. H. and Curtis, R. T. and Norton, S. P. and Parker, R. A. and Wilson, R. A.},
      TITLE = {Atlas of Finite Groups},
      PUBLISHER = {Oxford University Press},
      ADDRESS = {Eynsham},
      YEAR = {1985},
      PAGES = {xxxiv+252},
      ISBN = {0-19-853199-0},
      MRCLASS = {20D05 (20-02)},
      MRNUMBER = {0827219},
      MRREVIEWER = {R. L. Griess},
      ZBLNUMBER = {0568.20001},
      }
  • [MAtlwww] Go to document R. Wilson and R. et al., The Modular Atlas homepage.
    @misc{MAtlwww,
      author={Wilson, R. and \relax{et al.}},
      TITLE={The {M}odular {A}tlas homepage},
      URL = {http://www.math.rwth-aachen.de/~MOC/},
     }
  • [Da66] Go to document E. C. Dade, "Blocks with cyclic defect groups," Ann. of Math., vol. 84, pp. 20-48, 1966.
    @article {Da66, MRKEY = {0200355},
      AUTHOR = {Dade, E. C.},
      TITLE = {Blocks with cyclic defect groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {84},
      YEAR = {1966},
      PAGES = {20--48},
      ISSN = {0003-486X},
      MRCLASS = {20.80},
      MRNUMBER = {0200355},
      MRREVIEWER = {C. W. Curtis},
      ZBLNUMBER = {0163.27202},
      DOI = {10.2307/1970529},
     }
  • [DeMi] Go to document D. I. Deriziotis and G. O. Michler, "Character table and blocks of finite simple triality groups $^3D_4(q)$," Trans. Amer. Math. Soc., vol. 303, iss. 1, pp. 39-70, 1987.
    @article {DeMi, MRKEY = {0896007},
      AUTHOR = {Deriziotis, D. I. and Michler, G. O.},
      TITLE = {Character table and blocks of finite simple triality groups {$\sp 3D\sb 4(q)$}},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {303},
      YEAR = {1987},
      NUMBER = {1},
      PAGES = {39--70},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {20C15 (20C20 20G40)},
      MRNUMBER = {0896007},
      MRREVIEWER = {R. W. Carter},
      DOI = {10.2307/2000778},
      ZBLNUMBER = {0628.20014},
      }
  • [DM91] Go to document F. Digne and J. Michel, Representations of Finite Groups of Lie Type, Cambridge: Cambridge Univ. Press, 1991, vol. 21.
    @book {DM91, MRKEY = {1118841},
      AUTHOR = {Digne, Fran{ç}ois and Michel, Jean},
      TITLE = {Representations of Finite Groups of {L}ie Type},
      SERIES = {London Math. Soc. Student Texts},
      VOLUME = {21},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1991},
      PAGES = {iv+159},
      ISBN = {0-521-40117-8; 0-521-40648-X},
      MRCLASS = {20G05 (20-02 20C33 20G40)},
      MRNUMBER = {1118841},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0815.20014},
      DOI = {10.1017/CBO9781139172417},
     }
  • [En00] Go to document M. Enguehard, "Sur les $l$-blocs unipotents des groupes réductifs finis quand $l$ est mauvais," J. Algebra, vol. 230, iss. 2, pp. 334-377, 2000.
    @article {En00, MRKEY = {1775796},
      AUTHOR = {Enguehard, Michel},
      TITLE = {Sur les {$l$}-blocs unipotents des groupes réductifs finis quand {$l$} est mauvais},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {230},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {334--377},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G05 (20C20)},
      MRNUMBER = {1775796},
      MRREVIEWER = {Bhama Srinivasan},
      DOI = {10.1006/jabr.2000.8318},
      ZBLNUMBER = {0964.20020},
      }
  • [En08] Go to document M. Enguehard, "Vers une décomposition de Jordan des blocs des groupes réductifs finis," J. Algebra, vol. 319, iss. 3, pp. 1035-1115, 2008.
    @article {En08, MRKEY = {2379092},
      AUTHOR = {Enguehard, Michel},
      TITLE = {Vers une décomposition de {J}ordan des blocs des groupes réductifs finis},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {319},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {1035--1115},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G40 (20G05)},
      MRNUMBER = {2379092},
      MRREVIEWER = {Marc Cabanes},
      DOI = {10.1016/j.jalgebra.2007.06.036},
      ZBLNUMBER = {1194.20048},
      }
  • [FS82] Go to document P. Fong and B. Srinivasan, "The blocks of finite general linear and unitary groups," Invent. Math., vol. 69, iss. 1, pp. 109-153, 1982.
    @article {FS82, MRKEY = {0671655},
      AUTHOR = {Fong, Paul and Srinivasan, Bhama},
      TITLE = {The blocks of finite general linear and unitary groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {69},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {109--153},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20C20 (20G05)},
      MRNUMBER = {0671655},
      MRREVIEWER = {P. Donovan},
      DOI = {10.1007/BF01389188},
      ZBLNUMBER = {0507.20007},
      }
  • [FS89] Go to document P. Fong and B. Srinivasan, "The blocks of finite classical groups," J. Reine Angew. Math., vol. 396, pp. 122-191, 1989.
    @article {FS89, MRKEY = {0988550},
      AUTHOR = {Fong, Paul and Srinivasan, Bhama},
      TITLE = {The blocks of finite classical groups},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {396},
      YEAR = {1989},
      PAGES = {122--191},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {20G40 (20C20)},
      MRNUMBER = {0988550},
      MRREVIEWER = {Toshiaki Shoji},
      ZBLNUMBER = {0656.20039},
      DOI = {10.1515/crll.1989.396.122},
      }
  • [Ge93] Go to document M. Geck, "Basic sets of Brauer characters of finite groups of Lie type. II," J. London Math. Soc., vol. 47, iss. 2, pp. 255-268, 1993.
    @article {Ge93, MRKEY = {1207947},
      AUTHOR = {Geck, Meinolf},
      TITLE = {Basic sets of {B}rauer characters of finite groups of {L}ie type. {II}},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {47},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {255--268},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {20C33},
      MRNUMBER = {1207947},
      MRREVIEWER = {Fran{ç}ois Digne},
      DOI = {10.1112/jlms/s2-47.2.255},
      ZBLNUMBER = {0797.20013},
      }
  • [GlWo] Go to document D. Gluck and T. R. Wolf, "Brauer’s height conjecture for $p$-solvable groups," Trans. Amer. Math. Soc., vol. 282, iss. 1, pp. 137-152, 1984.
    @article {GlWo, MRKEY = {0728707},
      AUTHOR = {Gluck, David and Wolf, Thomas R.},
      TITLE = {Brauer's height conjecture for {$p$}-solvable groups},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {282},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {137--152},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {20C20},
      MRNUMBER = {0728707},
      MRREVIEWER = {Peter W. Donovan},
      DOI = {10.2307/1999582},
      ZBLNUMBER = {0543.20007},
      }
  • [Gr10] Go to document J. Gramain, "On a conjecture of G. Malle and G. Navarro on nilpotent blocks," Electron. J. Combin., vol. 18, iss. 1, p. 217, 2011.
    @article {Gr10, MRKEY = {2853074},
      AUTHOR = {Gramain, Jean-Baptiste},
      TITLE = {On a conjecture of {G}. {M}alle and {G}. {N}avarro on nilpotent blocks},
      JOURNAL = {Electron. J. Combin.},
      FJOURNAL = {Electronic Journal of Combinatorics},
      VOLUME = {18},
      YEAR = {2011},
      NUMBER = {1},
      PAGES = {Paper 217, 14},
      ISSN = {1077-8926},
      MRCLASS = {20C30 (20C20)},
      MRNUMBER = {2853074},
      MRREVIEWER = {Christopher P. Bendel},
      URL = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p217/pdf},
     }
  • [Hi] G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, 1990.
    @misc{Hi,
      author={Hi\ss, G.},
      TITLE={Zerlegungszahlen endlicher {G}ruppen vom {L}ie-{T}yp in nicht-definierender {C}harakteristik},
      YEAR={1990},
      NOTE={Habilitationsschrift, Aachen},
      ZBLNUMBER = {0832.20021},
     }
  • [How] Go to document R. B. Howlett, "Normalizers of parabolic subgroups of reflection groups," J. London Math. Soc., vol. 21, iss. 1, pp. 62-80, 1980.
    @article {How, MRKEY = {0576184},
      AUTHOR = {Howlett, Robert B.},
      TITLE = {Normalizers of parabolic subgroups of reflection groups},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {The Journal of the London Mathematical Society. Second Series},
      VOLUME = {21},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {62--80},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {20H15},
      MRNUMBER = {0576184},
      MRREVIEWER = {E. W. Ellers},
      DOI = {10.1112/jlms/s2-21.1.62},
      ZBLNUMBER = {0427.20040},
      }
  • [Hum] Go to document J. E. Humphreys, "Defect groups for finite groups of Lie type," Math. Z., vol. 119, pp. 149-152, 1971.
    @article {Hum, MRKEY = {0285623},
      AUTHOR = {Humphreys, James E.},
      TITLE = {Defect groups for finite groups of {L}ie type},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {119},
      YEAR = {1971},
      PAGES = {149--152},
      ISSN = {0025-5874},
      MRCLASS = {20.70},
      MRNUMBER = {0285623},
      MRREVIEWER = {J. Sancho San Rom{á}n},
      ZBLNUMBER = {0198.04502},
      DOI = {10.1007/BF01109967},
     }
  • [Ke] Go to document R. Kessar, "The Solomon system $\mathcal{F}_{ Sol}(3)$ does not occur as fusion system of a 2-block," J. Algebra, vol. 296, iss. 2, pp. 409-425, 2006.
    @article {Ke, MRKEY = {2201049},
      AUTHOR = {Kessar, Radha},
      TITLE = {The {S}olomon system {$\mathcal{F}_{\rm Sol}(3)$} does not occur as fusion system of a 2-block},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {296},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {409--425},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20D15 (20C20 20D05)},
      MRNUMBER = {2201049},
      MRREVIEWER = {Josu Sangroniz},
      DOI = {10.1016/j.jalgebra.2005.09.026},
      ZBLNUMBER = {1092.20010},
      }
  • [KKL] Go to document R. Kessar, S. Koshitani, and M. Linckelmann, "Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8," J. Reine Angew. Math., vol. 671, pp. 85-130, 2012.
    @article {KKL, MRKEY = {2983198},
      AUTHOR = {Kessar, Radha and Koshitani, Shigeo and Linckelmann, Markus},
      TITLE = {Conjectures of {A}lperin and {B}roué for 2-blocks with elementary abelian defect groups of order 8},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {671},
      YEAR = {2012},
      PAGES = {85--130},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {20Kxx},
      MRNUMBER = {2983198},
      ZBLNUMBER = {06100515},
      DOI = {10.1515/CRELLE.2011.162},
      }
  • [La78] Go to document P. Landrock, "The non-principal $2$-blocks of sporadic simple groups," Comm. Algebra, vol. 6, iss. 18, pp. 1865-1891, 1978.
    @article {La78, MRKEY = {0511167},
      AUTHOR = {Landrock, Peter},
      TITLE = {The non-principal {$2$}-blocks of sporadic simple groups},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {6},
      YEAR = {1978},
      NUMBER = {18},
      PAGES = {1865--1891},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {20D08},
      MRNUMBER = {0511167},
      MRREVIEWER = {Lino Di Martino},
      DOI = {10.1080/00927877808822326},
      ZBLNUMBER = {0387.20010},
      }
  • [Lu84] G. Lusztig, Characters of Reductive Groups over a Finite Field, Princeton, NJ: Princeton Univ. Press, 1984, vol. 107.
    @book {Lu84, MRKEY = {0742472},
      AUTHOR = {Lusztig, George},
      TITLE = {Characters of Reductive Groups over a Finite Field},
      SERIES = {Ann. of Math. Studies},
      VOLUME = {107},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1984},
      PAGES = {xxi+384},
      ISBN = {0-691-08350-9; 0-691-08351-7},
      MRCLASS = {20G05 (14L20 20C15)},
      MRNUMBER = {0742472},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0556.20033},
      }
  • [Lu88] G. Lusztig, "On the representations of reductive groups with disconnected centre," in Orbites Unipotentes et Représentations, I, Paris: Soc. Math. France, 1988, vol. 168, p. 10, 157-166.
    @incollection {Lu88, MRKEY = {1021495},
      AUTHOR = {Lusztig, George},
      TITLE = {On the representations of reductive groups with disconnected centre},
      BOOKTITLE = {Orbites Unipotentes et Repr{é}sentations, I},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      VOLUME = {168},
      YEAR = {1988},
      PAGES = {10, 157--166},
      ISSN = {0303-1179},
      MRCLASS = {20G05 (20G40)},
      MRNUMBER = {1021495},
      MRREVIEWER = {Robert B{é}dard},
      ZBLNUMBER = {0703.20036},
     }
  • [MaF] Go to document G. Malle, "Die unipotenten Charaktere von ${}^2F_4(q^2)$," Comm. Algebra, vol. 18, iss. 7, pp. 2361-2381, 1990.
    @article {MaF, MRKEY = {1063140},
      AUTHOR = {Malle, Gunter},
      TITLE = {Die unipotenten {C}haraktere von {${}\sp 2F\sb 4(q\sp 2)$}},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {18},
      YEAR = {1990},
      NUMBER = {7},
      PAGES = {2361--2381},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {20C15 (20C20 20G40)},
      MRNUMBER = {1063140},
      MRREVIEWER = {Richard Dipper},
      DOI = {10.1080/00927879008824026},
      ZBLNUMBER = {0721.20008},
      }
  • [MaH] Go to document G. Malle, "Height 0 characters of finite groups of Lie type," Represent. Theory, vol. 11, pp. 192-220, 2007.
    @article {MaH, MRKEY = {2365640},
      AUTHOR = {Malle, Gunter},
      TITLE = {Height 0 characters of finite groups of {L}ie type},
      JOURNAL = {Represent. Theory},
      FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society},
      VOLUME = {11},
      YEAR = {2007},
      PAGES = {192--220},
      ISSN = {1088-4165},
      MRCLASS = {20C33 (20G40)},
      MRNUMBER = {2365640},
      MRREVIEWER = {Gerhard Hiss},
      DOI = {10.1090/S1088-4165-07-00312-3},
      ZBLNUMBER = {1139.20008},
      }
  • [MN] Go to document G. Malle and G. Navarro, "Blocks with equal height zero degrees," Trans. Amer. Math. Soc., vol. 363, iss. 12, pp. 6647-6669, 2011.
    @article {MN, MRKEY = {2833571},
      AUTHOR = {Malle, Gunter and Navarro, Gabriel},
      TITLE = {Blocks with equal height zero degrees},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {363},
      YEAR = {2011},
      NUMBER = {12},
      PAGES = {6647--6669},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {20C15 (20C30 20C33)},
      MRNUMBER = {2833571},
      MRREVIEWER = {Donald L. White},
      DOI = {10.1090/S0002-9947-2011-05333-X},
      ZBLNUMBER = {05987998},
      }
  • [MT] Go to document G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge: Cambridge Univ. Press, 2011, vol. 133.
    @book {MT, MRKEY = {2850737},
      AUTHOR = {Malle, Gunter and Testerman, Donna},
      TITLE = {Linear Algebraic Groups and Finite Groups of {L}ie Type},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {133},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2011},
      PAGES = {xiv+309},
      ISBN = {978-1-107-00854-0},
      MRCLASS = {20G15 (20D06 20G05)},
      MRNUMBER = {2850737},
      MRREVIEWER = {Timothy C. Burness},
      ZBLNUMBER = {05852779},
      DOI = {10.1017/CBO9780511994777},
     }
  • [MChev] Go to document J. Michel, The GAP-part of the Chevie system.
    @misc{MChev,
      author={Michel, J.},
      TITLE={The {GAP}-part of the {C}hevie system},
      URL={ http://people.math.jussieu.fr/~jmichel/chevie/chevie.html},
     }
  • [Mue10] J. Müller, , 2010.
    @misc{Mue10,
      author={Müller, J.},
      NOTE={private communication},
      YEAR={2010},
     }
  • [Mu] Go to document M. Murai, "Block induction, normal subgroups and characters of height zero," Osaka J. Math., vol. 31, iss. 1, pp. 9-25, 1994.
    @article {Mu, MRKEY = {1262786},
      AUTHOR = {Murai, Masafumi},
      TITLE = {Block induction, normal subgroups and characters of height zero},
      JOURNAL = {Osaka J. Math.},
      FJOURNAL = {Osaka Journal of Mathematics},
      VOLUME = {31},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {9--25},
      ISSN = {0030-6126},
      CODEN = {OJMAA7},
      MRCLASS = {20C20},
      MRNUMBER = {1262786},
      URL = {http://projecteuclid.org/euclid.ojm/1200785085},
      ZBLNUMBER = {0830.20008},
      }
  • [NTs] H. Nagao and Y. Tsushima, Representations of Finite Groups, Boston, MA: Academic Press, 1989.
    @book {NTs, MRKEY = {0998775},
      AUTHOR = {Nagao, Hirosi and Tsushima, Yukio},
      TITLE = {Representations of Finite Groups},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1989},
      PAGES = {xviii+424},
      ISBN = {0-12-513660-9},
      MRCLASS = {20Cxx (20-01 20C20)},
      MRNUMBER = {0998775},
      MRREVIEWER = {Roderick Gow},
      ZBLNUMBER = {0673.20002},
      }
  • [NS12] G. Navarro and B. Späth, On Brauer’s height zero conjecture.
    @misc{NS12,
      author={Navarro, Gabriel and Späth, B.},
      TITLE={On {B}rauer's height zero conjecture},
      NOTE={\emph{J. Europ. Math. Soc.},
      to appear},
     }
  • [NT10] Go to document G. Navarro and P. H. Tiep, "Brauer’s height zero conjecture for the 2-blocks of maximal defect," J. Reine Angew. Math., vol. 669, pp. 225-247, 2012.
    @article {NT10, MRKEY = {2980457},
      AUTHOR = {Navarro, Gabriel and Tiep, Pham Huu},
      TITLE = {Brauer's height zero conjecture for the 2-blocks of maximal defect},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {669},
      YEAR = {2012},
      PAGES = {225--247},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {20C20},
      MRNUMBER = {2980457},
      ZBLNUMBER = {06081364},
      DOI = {10.1515/CRELLE.2011.147},
     }
  • [Ol90] Go to document J. B. Olsson, "On the $p$-blocks of symmetric and alternating groups and their covering groups," J. Algebra, vol. 128, iss. 1, pp. 188-213, 1990.
    @article {Ol90, MRKEY = {1031917},
      AUTHOR = {Olsson, Jørn B.},
      TITLE = {On the {$p$}-blocks of symmetric and alternating groups and their covering groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {128},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {188--213},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20C20 (05A15 05A17 20B30)},
      MRNUMBER = {1031917},
      MRREVIEWER = {Burkhard K{ü}lshammer},
      DOI = {10.1016/0021-8693(90)90049-T},
      ZBLNUMBER = {0695.20011},
      }
  • [Pu] L. Puig, On the Local Structure of Morita and Rickard Equivalences Between Brauer Blocks, Basel: Birkhäuser, 1999, vol. 178.
    @book {Pu, MRKEY = {1707300},
      AUTHOR = {Puig, Llu{\'ı}s},
      TITLE = {On the Local Structure of {M}orita and {R}ickard Equivalences Between {B}rauer Blocks},
      SERIES = {Progr. Math.},
      VOLUME = {178},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1999},
      PAGES = {vi+261},
      ISBN = {3-7643-6156-5},
      MRCLASS = {20C11 (16D90)},
      MRNUMBER = {1707300},
      MRREVIEWER = {Jeremy Rickard},
      ZBLNUMBER = {0929.20012},
      }
  • [Sch] K. Schewe, Blöcke Exzeptioneller Chevalley-Gruppen, Bonn: Universität Bonn Mathematisches Institut, 1985, vol. 165.
    @book {Sch, MRKEY = {0825237},
      AUTHOR = {Schewe, Klaus-Dieter},
      TITLE = {Blöcke Exzeptioneller {C}hevalley-{G}ruppen},
      SERIES = {Bonner Mathematische Schriften [Bonn Math. Publ.]},
      VOLUME={165},
      NOTE = {Dissertation, Rheinische Friedrich-Wilhelm-Universit{ä}t, Bonn, 1985},
      PUBLISHER = {Universität Bonn Mathematisches Institut},
      ADDRESS = {Bonn},
      YEAR = {1985},
      PAGES = {ii+196},
      MRCLASS = {20G05 (20C20)},
      MRNUMBER = {0825237},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0644.20027},
     }
  • [The95] J. Thévenaz, $G$-Algebras and Modular Representation Theory, New York: The Clarendon Press Oxford University Press, 1995.
    @book {The95, MRKEY = {1365077},
      AUTHOR = {Th{é}venaz, Jacques},
      TITLE = {{$G$}-Algebras and Modular Representation Theory},
      SERIES = {Oxford Math. Monogr.},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {xxviii+470},
      ISBN = {0-19-853587-2},
      MRCLASS = {20C20 (20-02 20C05)},
      MRNUMBER = {1365077},
      MRREVIEWER = {Burkhard K{ü}lshammer},
      ZBLNUMBER = {0837.20015},
      }
  • [Wa66] Go to document H. N. Ward, "On Ree’s series of simple groups," Trans. Amer. Math. Soc., vol. 121, pp. 62-89, 1966.
    @article {Wa66, MRKEY = {0197587},
      AUTHOR = {Ward, Harold N.},
      TITLE = {On {R}ee's series of simple groups},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {121},
      YEAR = {1966},
      PAGES = {62--89},
      ISSN = {0002-9947},
      MRCLASS = {20.80 (20.29)},
      MRNUMBER = {0197587},
      MRREVIEWER = {Rimhak Ree},
      ZBLNUMBER = {0139.24902},
      DOI = {10.1090/S0002-9947-1966-0197587-8},
     }

Authors

Radha Kessar

Department of Mathematics
City University London
Northampton Square
London EC1V 0HB
United Kingdom

Gunter Malle

Fachbereich Mathematik
TU Kaiserslautern
Postfach 3049
67653 Kaisersautern
Germany.