Blocks with equal height zero degrees

Malle, Gunter; Navarro, Gabriel

doi:10.1090/S0002-9947-2011-05333-X

Blocks with equal height zero degrees
HTML articles powered by AMS MathViewer

by Gunter Malle and Gabriel Navarro

Trans. Amer. Math. Soc. 363 (2011), 6647-6669

DOI: https://doi.org/10.1090/S0002-9947-2011-05333-X

Published electronically: June 15, 2011

PDF | Request permission

Abstract:

We investigate a natural class of blocks of finite groups: the blocks such that all of their height zero characters have the same degree. It is conceivable that these blocks, which are globally defined, are exactly the Broué-Puig (locally defined) nilpotent blocks and we offer some partial results in this direction. The most difficult result here is to prove that, with one family of possible exceptions, blocks with equal height zero degrees of simple groups have abelian defect groups and are in fact nilpotent.

References

J. Alperin and Michel Broué, Local methods in block theory, Ann. of Math. (2) 110 (1979), no. 1, 143–157. MR 541333, DOI 10.2307/1971248
J. An, C. Eaton, Nilpotent blocks of quasisimple groups for odd primes. To appear in J. reine angew. Math., 2010.
J. An, C. Eaton, Nilpotent blocks of quasisimple groups for the prime 2. Preprint, 2009.
Cédric Bonnafé, Quasi-isolated elements in reductive groups, Comm. Algebra 33 (2005), no. 7, 2315–2337. MR 2153225, DOI 10.1081/AGB-200063602
Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
Michel Broué and Gunter Malle, Generalized Harish-Chandra theory, Representations of reductive groups, Publ. Newton Inst., vol. 16, Cambridge Univ. Press, Cambridge, 1998, pp. 85–103. MR 1714151, DOI 10.1017/CBO9780511600623.006
Michel Broué, Gunter Malle, and Jean Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235832
Michel Broué and Jean Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989), 56–67 (French). MR 983059, DOI 10.1515/crll.1989.395.56
Michel Broué and Lluís Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), no. 2, 117–128. MR 558864, DOI 10.1007/BF01392547
Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756, DOI 10.1017/CBO9780511542763
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
D. I. Deriziotis and G. O. Michler, Character table and blocks of finite simple triality groups $^3D_4(q)$, Trans. Amer. Math. Soc. 303 (1987), no. 1, 39–70. MR 896007, DOI 10.1090/S0002-9947-1987-0896007-9
Silvio Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360 (2008), no. 1, 135–152. MR 2341997, DOI 10.1090/S0002-9947-07-04155-4
Silvio Dolfi and Gabriel Navarro, Large orbits of elements centralized by a Sylow subgroup, Arch. Math. (Basel) 93 (2009), no. 4, 299–304. MR 2558521, DOI 10.1007/s00013-009-0016-5
Michel Enguehard, Sur les $l$-blocs unipotents des groupes réductifs finis quand $l$ est mauvais, J. Algebra 230 (2000), no. 2, 334–377 (French). MR 1775796, DOI 10.1006/jabr.2000.8318
Michel Enguehard, Vers une décomposition de Jordan des blocs des groupes réductifs finis, J. Algebra 319 (2008), no. 3, 1035–1115 (French, with English summary). MR 2379092, DOI 10.1016/j.jalgebra.2007.06.036
Paul Fong, On decomposition numbers of $J_{1}$ and $R(q)$, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani & Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 415–422. MR 0357578
Meinolf Geck, Lacrimioara Iancu, and Gunter Malle, Weights of Markov traces and generic degrees, Indag. Math. (N.S.) 11 (2000), no. 3, 379–397. MR 1813479, DOI 10.1016/S0019-3577(00)80005-1
Gerhard Hiss and Josephine Shamash, $3$-blocks and $3$-modular characters of $G_2(q)$, J. Algebra 131 (1990), no. 2, 371–387. MR 1058552, DOI 10.1016/0021-8693(90)90181-M
Gerhard Hiss and Josephine Shamash, $2$-blocks and $2$-modular characters of the Chevalley groups $G_2(q)$, Math. Comp. 59 (1992), no. 200, 645–672. MR 1134731, DOI 10.1090/S0025-5718-1992-1134731-9
James E. Humphreys, Defect groups for finite groups of Lie type, Math. Z. 119 (1971), 149–152. MR 285623, DOI 10.1007/BF01109967
I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
I. M. Isaacs and Stephen D. Smith, A note on groups of $p$-length $1$, J. Algebra 38 (1976), no. 2, 531–535. MR 393215, DOI 10.1016/0021-8693(76)90236-2
G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157–166. Orbites unipotentes et représentations, I. MR 1021495
Gunter Malle, Die unipotenten Charaktere von ${}^2F_4(q^2)$, Comm. Algebra 18 (1990), no. 7, 2361–2381 (German). MR 1063140, DOI 10.1080/00927879008824026
Gunter Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), no. 3, 768–826 (German, with German summary). MR 1358486, DOI 10.1006/jabr.1995.1329
Gunter Malle, Almost irreducible tensor squares, Comm. Algebra 27 (1999), no. 3, 1033–1051. MR 1669100, DOI 10.1080/00927879908826479
Gunter Malle, Height 0 characters of finite groups of Lie type, Represent. Theory 11 (2007), 192–220. MR 2365640, DOI 10.1090/S1088-4165-07-00312-3
G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
Gabriel Navarro, Nilpotent characters, Pacific J. Math. 169 (1995), no. 2, 343–351. MR 1346259
Gabriel Navarro, Brauer characters relative to a normal subgroup, Proc. London Math. Soc. (3) 81 (2000), no. 1, 55–71. MR 1757046, DOI 10.1112/S0024611500012351
Gabriel Navarro and Geoffrey R. Robinson, Blocks with $p$-power character degrees, Proc. Amer. Math. Soc. 133 (2005), no. 10, 2845–2851. MR 2159761, DOI 10.1090/S0002-9939-05-07915-3
Herbert Pahlings, Normal $p$-complements and irreducible characters, Math. Z. 154 (1977), no. 3, 243–246. MR 439920, DOI 10.1007/BF01214323
Jørn B. Olsson, Combinatorics and representations of finite groups, Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen [Lecture Notes in Mathematics at the University of Essen], vol. 20, Universität Essen, Fachbereich Mathematik, Essen, 1993. MR 1264418
Tetsuro Okuyama and Yukio Tsushima, Local properties of $p$-block algebras of finite groups, Osaka Math. J. 20 (1983), no. 1, 33–41. MR 695615
W. F. Reynolds, Blocks and normal subgroups of finite groups, Nagoya Math. J. 22 (1963), 15–32. MR 153729