Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T04:40:55.740Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Action of the Sporadic Simple Baby Monster Group on its Conjugacy Class 2B

Published online by Cambridge University Press:  01 February 2010

Jürgen Müller
Jürgen Müller
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany, [email protected]

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.

We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 21+22.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 15 - 27
Copyright
Copyright © London Mathematical Society 2008

References

1.Bannai, E. and Ito, T., Algebraic combinatorics I: association schemes (Benjamin, 1984).Google Scholar
2.Breuer, T., ‘Multiplicity-free permutation characters in GAP II’, Preprint, 2005, http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/.Google Scholar
3.Breuer, T. and Lux, K., ‘The multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups’, Comm. Algebra 24 (1996) 2293–2316; errata in http://www.math.rwth-aachen.de/~Thomas.Breuer/multfree/.CrossRefGoogle Scholar
4.Breuer, T. and Müller, J., ‘Character tables of endomorphism rings of multiplicity-free permutation modules of the sporadic simple groups and their cyclic and bicyclic extensions’, database, 2006, http://www.math.rwth-aachen.de/~Juergen.Mueller/.Google Scholar
5.Brouwer, A., Cohen, A. and Neumaier, A., Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 18 (Springer, 1989).CrossRefGoogle Scholar
6.Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R., Atlas of finite groups (Oxford University Press, 1985).Google Scholar
7.Davidoff, G., Sarnak, P. and Valette, A., Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts 55 (Cambridge University Press, 2003).Google Scholar
8.Feit, W., The representation theory of finite groups, North-Holland Mathematical Library 25 (North-Holland, 1982).Google Scholar
9.THE GAP GROUP, ‘GAP—Groups, Algorithms, and Programming’, version 4.4 (2006), http://www.gap-system.org.Google Scholar
10.Higman, D., ‘A monomial character of Fischer's Baby Monster’, Proc. of the conference on finite groups, Utah, 1975 (ed. Gross, F. and Scott, W.; Academic Press, 1976) 277–283.CrossRefGoogle Scholar
11.Holt, D., Eick, B. and O'Brien, E., Handbook of computational group theory, Discrete Mathematics and its Applications (Chapman & Hall, 2005).CrossRefGoogle Scholar
12.Jansen, C., ‘The minimal degrees of faithful representations of the sporadic simple groups and their covering groups’, LMS J. Gomput. Math. 8 (2005) 122–144.CrossRefGoogle Scholar
13.Jansen, C., Lux, K., Parker, R. and Wilson, R., An atlas of Brauer characters (Oxford University Press, 1995).Google Scholar
14.Ivanov, A., Linton, S., Lux, K., Saxl, J. and Soicher, L., ‘Distance-transitive representations of the sporadic groups’, Comm. Algebra 23 (1995) 3379–3427.CrossRefGoogle Scholar
15.Landrock, P., Finite group algebras and their modules, London Mathematical Society Lecture Note Series 84 (Cambridge University Press, 1983).CrossRefGoogle Scholar
16.Linton, S. and Mpono, Z., ‘Multiplicity-free permutation characters of covering groups of sporadic simple groups’, Preprint, 2001.Google Scholar
17.Lübeck, F. and Neunhöffer, M., ‘Enumerating large orbits and direct condensation’, Experiment. Math. 10 (2001) 197–205.CrossRefGoogle Scholar
18.Lux, K., Müller, J. and Ringe, M., ‘Peakwordcondensation and submodulelattices, an application of the MeatAxe’, J. Symb. Gomput. 17 (1994) 529–544.CrossRefGoogle Scholar
19.Lux, K. and Wiegelmann, M., ‘Determination of socle series using the condensation method’, Computational algebra and number theory, Milwaukee, 1996, J. Symb. Gomput. 31 (2001) 163–178.CrossRefGoogle Scholar
20.Müller, J., ‘On the multiplicity-free actions of the sporadic simple groups’, Preprint, 2007, http://www.math.rwth-aachen.de/~Juergen.Mueller/.Google Scholar
21.Müller, J., On endomorphism rings and character tables, Habilitations-schrift, RWTH Aachen, 2003, http://www.math.rwth-aachen.de/~Juergen.Mueller/.Google Scholar
22.Müller, J., Neunhöffer, M. and Noeske, F., GAP-4 package ORB, in preparation, http://www.math.rwth-aachen.de/~Juergen.Mueller/.Google Scholar
23.Müller, J., Neunhöffer, M. and Wilson, R., ‘Enumerating big orbits and an application, B acting on the cosets of Fi23’, J. Algebra 314 (2007) 75–96.CrossRefGoogle Scholar
24.Parker, R., private communication.Google Scholar
25.Ringe, M., ‘The C-MeatAxe’, version 2.4 (RWTH Aachen, 2007), http://www.math.rwth-aachen.de/~MTX/.Google Scholar
26.Rowley, P. and Walker, L., ‘A 11, 707, 448, 673, 375 vertex graph related to the baby monster I’, J. Combin. Theory Ser. A 107 (2004) 181–213.CrossRefGoogle Scholar
27.Rowley, P. and Walker, L., ‘A 11, 707, 448, 673, 375 vertex graph related to the baby monster II’, J. Combin. Theory Ser. A 107 (2004) 215–261.CrossRefGoogle Scholar
28.Schur, I., ‘Zur Theorie der einfach transitiven Permutationsgruppen’, Sitzungsberichte der Preußischen Akademie der Wissenschaften (1933) 598–623.Google Scholar
29.Wilson, R., ‘Standard generators for sporadic simple groups’, J. Algebra 184 (1996) 505–515.CrossRefGoogle Scholar
30.Wilson, R., ‘A new construction of the Baby Monster and its applications’, Bull. London Math. Soc. 25 (1993) 431–437.CrossRefGoogle Scholar
31.Wilson, R., Parker, R., Nickerson, S. and Bray, J., ‘Atlas of finite group representations’, database, 2005, http://brauer.maths.qmul.ac.uk/Atlas/.Google Scholar
32.Zieschang, P., An algebraic approach to association schemes, Lecture Notes in Mathematics 1628 (Springer, 1996).CrossRefGoogle Scholar