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On Loewy lengths of blocks

Published online by Cambridge University Press:  20 February 2014

SHIGEO KOSHITANI
Affiliation:
Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522Japan. e-mail: koshitan@math.s.chiba-u.ac.jp
BURKHARD KÜLSHAMMER
Affiliation:
Mathematisches Institut, Friedrich–Schiller–Universität, D-07737 Jena, Germany. e-mail: kuelshammer@uni-jena.de
BENJAMIN SAMBALE
Affiliation:
Mathematisches Institut, Friedrich–Schiller–Universität, D-07737 Jena, Germany. e-mail: benjamin.sambale@uni-jena.de

Abstract

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Alperin, J. L., Collins, M. J. and Sibley, D. A.Projective modules, filtrations and Cartan invariants. Bull. London Math. Soc. 16 (1984), 416420.Google Scholar
[2]Benson, D. J.The Loewy structure of the projective indecomposable modules for A 8 in characteristic 2. Comm. Algebra 11 (1983), 13951432.Google Scholar
[3]Benson, D. J.The Loewy structure of the projective indecomposable modules for A 9 in characteristic 2. Comm. Algebra 11 (1983), 14331453.Google Scholar
[4]Bonnafé, C.Representations of ${\rm SL}_2(\mathbb F_q)$. Algebr. Appl. vol. 13 (Springer-Verlag, London, 2011).Google Scholar
[5]Brauer, R. and Nesbitt, C.On the modular characters of groups. Ann. of Math. (2) 42 (1941), 556590.Google Scholar
[6]Broué, M.Brauer coefficients of p-subgroups associated with a p-block of a finite group. J. Algebra 56 (1979), 365383.Google Scholar
[7]Chang, B. and Ree, R.The characters of G 2(q). In: Symposia Mathematica, vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), 395–413. (Academic Press, London, 1974).Google Scholar
[8]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.ATLAS of Finite Groups (Oxford University Press, 1985).Google Scholar
[9]Danz, S. 3-blocks of weight 3, private communication (Nov 8, 2012).Google Scholar
[10]Eaton, C. W., Kessar, R., Külshammer, B. and Sambale, B.2-blocks with abelian defect groups. Adv. Math. 254 (2014), 706735.Google Scholar
[11]Fong, P. and Srinivasan, B.Blocks with cyclic defect groups in GL(n, q). Bull. Amer. Math. Soc. (N.S.) 3 (1980), 10411044.Google Scholar
[12]The GAP Group GAP – Groups, Algorithms and Programming, Version 4.6.4 (2013) (http://www.gap-system.org).Google Scholar
[13]Gorenstein, D., Lyons, R. and Solomon, R.The classification of the finite simple groups, Math. Surveys Monogr., vol. 40.I (American Mathematical Society, Providence, RI, 1994).Google Scholar
[14]Gorenstein, D., Lyons, R. and Solomon, R.The classification of the finite simple groups. no 3. Part I. Chapter A. Math. Surveys Monogr. vol. 40 (American Mathematical Society, Providence, RI, 1998).Google Scholar
[15]Hiss, G. and Lux, K.Brauer Trees of Sporadic Groups. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1989).Google Scholar
[16]Hiss, G. and Shamash, J.3-blocks and 3-modular characters of G 2(q). J. Algebra 131 (1990), 371387.Google Scholar
[17]Hu, Y. and Ye, J.On the first Cartan invariant for the finite group of type G 2. Comm. Algebra 30 (2002), 45494573.Google Scholar
[18]Humphreys, J. E.Modular representations of finite groups of Lie type. London Math. Society Lecture Note Series vol. 326 (Cambridge University Press, Cambridge, 2006).Google Scholar
[19]Huppert, B.Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134 (Springer-Verlag, Berlin, 1967).Google Scholar
[20]James, G.The decomposition matrices of GLn(q) for n ≤ 10. Proc. London Math. Soc. (3) 60 (1990), 225265.Google Scholar
[21]James, G. and Kerber, A.The representation theory of the symmetric group. Encyclopedia Math. Appl. vol. 16 (Addison-Wesley Publishing Co., Reading, Mass., 1981).Google Scholar
[22]Jennings, S. A.The structure of the group ring of a p-group over a modular field. Trans. Amer. Math. Soc. 50 (1941), 175185.Google Scholar
[23]Kessar, R. and Linckelmann, M.On blocks with Frobenius inertial quotient. J. Algebra 249 (2002), 127146.Google Scholar
[24]Koshitani, S.On lower bounds for the radical of a block ideal in a finite p-solvable group. Proc. Edinburgh Math. Soc. (2) 27 (1984), 6571.Google Scholar
[25]Koshitani, S.Cartan invariants of group algebras of finite groups. Proc. Amer. Math. Soc. 124 (1996), 23192323.Google Scholar
[26]Koshitani, S. On the projective cover of the trivial module over a group algebra of a finite group. Comm. Algebra (to appear).Google Scholar
[27]Koshitani, S. and Miyachi, H.Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian Sylow 3-subgroup of order 9. Comm. Algebra 29 (2001), 45094522.CrossRefGoogle Scholar
[28]Koshitani, S. and Yoshii, Y.Eigenvalues of Cartan matrices of principal 3-blocks of finite groups with abelian Sylow 3-subgroups. J. Algebra 324 (2010), 19851993.Google Scholar
[29]Külshammer, B.Bemerkungen über die Gruppenalgebra als symmetrische Algebra. II. J. Algebra 75 (1982), 5969.Google Scholar
[30]Külshammer, B.Crossed products and blocks with normal defect groups. Comm. Algebra 13 (1985), 147168.Google Scholar
[31]Külshammer, B.Group-theoretical descriptions of ring-theoretical invariants of group algebras. In Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 425–442 Progr. Math., vol. 95 (Birkhäuser, Basel, 1991).Google Scholar
[32]Müller, J.The Monster in characteristic 11. Private communication (May 2, 2013).Google Scholar
[33]Naehrig, M. Die Brauer–Büme des Monsters M in Charakteristik 29, Diplomarbeit, 2002, Aachen.Google Scholar
[34]Narasaki, R. and Uno, K.Isometries and extra special Sylow groups of order p 3. J. Algebra 322 (2009), 20272068.Google Scholar
[35]Neusel, M. D. and Smith, L.Invariant theory of finite groups. Math. Surveys Monogr., vol. 94 (American Mathematical Society, Providence, RI, 2002).Google Scholar
[36]Okuyama, T.On blocks of finite groups with radical cube zero. Osaka J. Math. 23 (1986), 461465.Google Scholar
[37]Oppermann, S.A lower bound for the representation dimension of kCnp. Math. Z. 256 (2007), 481490.Google Scholar
[38]Puig, L.Nilpotent blocks and their source algebras. Invent. Math. 93 (1988), 77116.Google Scholar
[39]Sambale, B.Fusion systems on metacyclic 2-groups. Osaka J. Math. 49 (2012), 325329.Google Scholar
[40]Schreier, O.Über die Erweiterung von Gruppen II. Abhandlungen Hamburg 4 (1926), 321346.Google Scholar
[41]Scopes, J.Symmetric group blocks of defect two. Quart. J. Math. Oxford Ser. (2) 46 (1995), 201234.Google Scholar
[42]Tan, K. M.Martin's conjecture holds for weight 3 blocks of symmetric groups. J. Algebra 320 (2008), 11151132.Google Scholar
[43]Webb, P. J.The Auslander-Reiten quiver of a finite group. Math. Z. 179 (1982), 97121.Google Scholar
[44]White, D. L.Decomposition numbers of Sp(4,q) for primes dividing q ± 1. J. Algebra 132 (1990), 488500.Google Scholar
[45]White, D. L.Decomposition numbers of Sp4(2a) in odd characteristics. J. Algebra 177 (1995), 264276.Google Scholar
[46]Wilkinson, D.The groups of exponent p and order p 7 (p any prime). J. Algebra 118 (1988), 109119.Google Scholar
[47]Wilson, R. A.The finite simple groups. Graduate Texts in Math., vol. 251 (Springer-Verlag London Ltd., London, 2009).Google Scholar
[48]Winter, D. L.The automorphism group of an extraspecial p-group. Rocky Mountain J. Math. 2 (1972), 159168.Google Scholar