Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T22:55:08.915Z Has data issue: false hasContentIssue false
  • English
  • Français

The Alperin weight conjecture and Uno’s conjecture for the Monster 𝕄, p odd

Part of: Representation theory of groups

Published online by Cambridge University Press:  27 August 2010

Jianbei An  and
R. A. Wilson
Jianbei An
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand (email: an@math.auckland.ac.nz)
R. A. Wilson
Affiliation:
Department of Mathematics, The University of Birmingham, Birmingham B15 2TT, United Kingdom (email: R.A.Wilson@bham.ac.uk)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that p is 3,5,7,11 or 13. We classify the radical p-chains of the Monster 𝕄 and verify the Alperin weight conjecture and Uno’s reductive conjecture for 𝕄, the latter being a refinement of Dade’s reductive conjecture and the Isaacs–Navarro conjecture.

MSC classification

Secondary: 20C20: Modular representations and characters 20C34: Representations of sporadic groups 20D08: Simple groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 320 - 356
Copyright
Copyright © London Mathematical Society 2010

References

[1] Alperin, J. L., ‘Weights for finite groups’, The Arcata conference on representations of finite groups, Proceedings of Symposia in Pure Mathematics 47 (American Mathematical Society, Providence, RI, 1987) 369–379.CrossRefGoogle Scholar
[2] An, J., ‘The Alperin and Dade conjectures for the simple Held group’, J. Algebra 189 (1997) 34–57.CrossRefGoogle Scholar
[3] An, J., ‘Dade’s invariant conjecture for the Chevalley groups G 2(q) in the defining characteristic, q=2a,3a’, Algebra Colloq. 10 (2003) 519–533.Google Scholar
[4] An, J., ‘Uno’s invariant conjecture for the general linear and unitary groups in nondefining characteristics’, J. Algebra 284 (2005) 462–479.CrossRefGoogle Scholar
[5] An, J., Cannon, J., O’Brien, E. A. and Unger, W., ‘The Alperin and Uno conjectures for the Fischer simple group  Fi′24’, LMS J. Comput. Math. 11 (2008) 100–145.CrossRefGoogle Scholar
[6] An, J. and Eaton, C. W., ‘The p-local rank of a block’, J. Group Theory 3 (2000) 369–380.CrossRefGoogle Scholar
[7] An, J. and Eaton, C. W., ‘Modular representation theory of blocks with trivial intersection defect groups’, Algebr. Represent. Theory 8 (2005) 427–448.CrossRefGoogle Scholar
[8] An, J. and O’Brien, E. A., ‘A local strategy to decide the Alperin and Dade conjectures’, J. Algebra 206 (1998) 183–207.CrossRefGoogle Scholar
[9] An, J. and O’Brien, E. A., ‘The Alperin and Dade conjectures for the simple Fischer group  Fi23’, Internat. J. Algebra Comput. 9 (1999) 621–670.CrossRefGoogle Scholar
[10] An, J. and O’Brien, E. A., ‘Conjectures on the character degrees of the Harada–Norton simple group HN’, Israel J. Math. 137 (2003) 157–181.CrossRefGoogle Scholar
[11] An, J. and Wilson, R. A., ‘The Alperin and Uno conjectures for the Baby Monster 𝔹, p odd’, LMS J. Comput. Math. 7 (2004) 120–166.CrossRefGoogle Scholar
[12] Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: T the user language’, J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
[13] Bray, J. N. and Wilson, R. A., ‘Explicit representations of maximal subgroups of the Monster’, J. Algebra 300 (2006) 834–857.CrossRefGoogle Scholar
[14] Burgoyne, N. and Williamson, C., ‘On a theorem of Borel and Tits for finite Chevalley groups’, Arch. Math. 27 (1976) 489–491.CrossRefGoogle Scholar
[15] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
[16] Dade, E. C., ‘Counting characters in blocks I’, Invent. Math. 109 (1992) 187–210.CrossRefGoogle Scholar
[17] Dade, E. C., ‘Counting characters in blocks, II.9.’, Representation theory of finite groups (Proceedings of a special research quarter at the Ohio State University, spring, 1995), Ohio State University Mathematical Research Institute Publications 6 (de Gruyter, Berlin, 1997) 45–59.CrossRefGoogle Scholar
[18] The GAP group, ‘GAP—groups, algorithms, and programming, version 4’. Lehrstuhl D für Mathematik, RWTH Aachen and School of Mathematical and Computational Sciences, University of St Andrews, 2000.Google Scholar
[19] Hiss, G. and Lux, K., Brauer trees of sporadic groups (Oxford University Press, Oxford, 1989).Google Scholar
[20] Isaacs, I. M. and Navarro, G., ‘New refinements of the McKay conjecture for arbitrary finite groups’, Ann of Math. (2) 156 (2002) 333–344.CrossRefGoogle Scholar
[21] Knörr, R., ‘On the vertices of irreducible modules’, Ann. of Math. (2) 110 (1979) 487–499.CrossRefGoogle Scholar
[22] Linton, S., Parker, R., Walsh, P. and Wilson, R., ‘Computer construction of the Monster’, J. Group Theory 1 (1998) 307–337.Google Scholar
[23] Uno, K., ‘Conjectures on character degrees for the simple Thompson group’, Osaka J. Math. 41 (2004) no. 1, 11–36.Google Scholar
[24] Wilson, R. A., ‘The odd-local subgroups of the Monster’, J. Austral. Math. Soc. 44 (1988) 1–16.CrossRefGoogle Scholar
[25] Wilson, R. A. et al., ‘ATLAS of finite group representations’, http://web.mat.bham.ac.uk/atlas.Google Scholar
[26] Yoshiara, S., ‘Odd radical subgroups of some sporadical simple groups’, J. Algebra 291 (2005) 90–107.CrossRefGoogle Scholar