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$ 2$-blocks and $ 2$-modular characters of the Chevalley groups $ G\sb 2(q)$


Authors: Gerhard Hiss and Josephine Shamash
Journal: Math. Comp. 59 (1992), 645-672
MSC: Primary 20C20; Secondary 20C33, 20C40
DOI: https://doi.org/10.1090/S0025-5718-1992-1134731-9
MathSciNet review: 1134731
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Abstract: We first determine the distribution of the ordinary irreducible characters of the exceptional Chevalley group $ {G_2}(q)$, q odd, into 2-blocks. This is done by using the method of central characters. Then all but two of the irreducible 2-modular characters are determined. The results are given in the form of decomposition matrices. The methods here involve concepts from modular representation theory and symbolic computations with the computer algebra system MAPLE. As a corollary, the smallest degree of a faithful representation of $ {G_2}(q)$, q odd, over a field of characteristic 2 is obtained.


References [Enhancements On Off] (What's this?)

  • [1] D. W. Burry, Components of induced modules, J. Algebra 87 (1984), 483-492. MR 739948 (85j:20006)
  • [2] R. W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley, New York, 1985. MR 794307 (87d:20060)
  • [3] B. Chang and R. Ree, The characters of $ {G_2}(q)$, Symposia Mathematica, vol. 13, Academic Press, London, 1974, pp. 395-413. MR 0364419 (51:673)
  • [4] H. Enomoto, The characters of $ {G_2}(q),q = {3^n}$, Japan. J. Math. 2 (1976), 191-248. MR 0437628 (55:10552)
  • [5] K. Erdmann, Algebras and semidihedral defect groups. I, Proc. London Math. Soc. 57 (1988), 109-150. MR 940432 (89e:20010)
  • [6] -, Algebras and semidihedral defect groups. II, Proc. London Math. Soc. 60 (1990), 123-165. MR 1023807 (91a:20009)
  • [7] G. Hiss, On the decomposition numbers of $ {G_2}(q)$, J. Algebra 120 (1989), 339-360. MR 989902 (90j:20011)
  • [8] G. Hiss and J. Shamash, 3-blocks and 3-modular characters of $ {G_2}(q)$, J. Algebra 131 (1990), 371-387. MR 1058552 (91h:20017)
  • [9] R. B. Howlett and R. W. Kilmoyer, Principal series representations of finite groups with split BN-pairs, Comm. Algebra 8 (1980), 543-583. MR 561752 (81d:20009)
  • [10] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia Math. Appl., vol. 16, Addison-Wesley, Reading, MA, 1981. MR 644144 (83k:20003)
  • [11] P. Landrock, Finite group algebras and their modules, London Math. Soc. Lecture Notes, vol. 84, Cambridge Univ. Press, Cambridge, 1983. MR 737910 (85h:20002)
  • [12] J. Shamash, Blocks and Brauer trees for groups of type $ {G_2}(q)$, The Arcata Conference on Representations of Finite Groups, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 283-295. MR 933418 (89c:20024)
  • [13] -, Brauer trees for blocks of cyclic defect in the groups $ {G_2}(q)$ for primes dividing $ {q^2} \pm q + 1$, J. Algebra 123 (1989), 378-396. MR 1000493 (90m:20016)
  • [14] -, Blocks and Brauer trees in the groups $ {G_2}(q)$ for primes dividing $ q \pm 1$, Comm. Algebra 17 (1989), 1901-1949. MR 1013474 (90j:20021)
  • [15] -, Blocks and Brauer trees for the groups $ {G_2}({2^k}),{G_2}({3^k})$, Comm. Algebra 20 (1992), 1375-1387.
  • [16] D. L. White, The 2-blocks and decomposition numbers of $ Sp(4,q)$, q odd, J. Algebra 131 (1990) 703-725. MR 1058575 (91g:20012)

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DOI: https://doi.org/10.1090/S0025-5718-1992-1134731-9
Article copyright: © Copyright 1992 American Mathematical Society

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