1Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. Email: acme@ifm.umich.mx
2Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. Email: valero@fismat.umich.mx
Alperins Weight Conjecture was originally formulated for algebraically closed fields (see cite [1]). For some families of groups --such as the symmetric groups-- it is known to hold for arbitrary fields (see cite [2]), so it is reasonable to ask whether this conjecture holds for arbitrary fields, and in particular, if it holds for finite fields. We wrote computer software in MAGMA (see cite [8]) to test Alperins Weight Conjecture for finite fields, and tested this software on groups of small order and the prime fields whose characteristics divide the order of the groups. We found no counterexamples to this version of Alperins Conjecture for groups of order up to 100.
Key words: Group representation, Alperin's conjecture, weight, software, computational.
La conjetura de pesos de Alperin fue formulada originalmente para campos algebraicamente cerrados. Para algunas familias de grupos --como por ejemplo los grupos simétricos-- esta Conjetura es válida para todos los campos, y en particular, para los campos finitos. Es razonable preguntar si dicha Conjetura permanecerá válida para todos los grupos y todos los campos, y en particular para los campos finitos. En este artículo verificamos (usando MAGMA) la Conjetura de Pesos de Alperin para todos los grupos de orden menor o igual a 100 y todos los campos primos cuyas características dividen el orden de cada grupo.
Palabras clave: Grupo de representaciones, conjetura de Alperin, peso, software, computacional.
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References
]]>[1] Alperin, J., Weights for finite groups, `The Arcata Conference on Representations of Finite Groups´, (1987), Proceedings of symposia in pure mathematics, American Mathematical Society, Providence, United States, p. 369-379. [ Links ]
[2] Alperin, J. & Fong, P., `Weights for symmetric and general linear groups´, Journal of Algebra 131, (1990), 2-22. [ Links ]
[3] An, J., `2 weights for general linear groups´, J. Algebra 149, (1992), 500-527. [ Links ]
[4] An, J., `2 weights for unitary groups´, Trans. Amer. Math. Soc 339, (1993a), 251-278. [ Links ]
[5] An, J., `Weights for the simple Ree groups 2g2(q2)´, New Zealand J. Math 22, (1993b), 1-8. [ Links ] ]]>
[6] An, J., `Weights for the Steinberg triality groups 3d4(q)´, Math Z. 218, (1995), 273-290. [ Links ]
[7] An, J. & Conder, M., `The Alperin and Dade conjectures for the simple Mathieu groups´, Comm. Algebra 23, (1995), 2797-2823. [ Links ]
[8] Bosma, W., Cannon, J. & Playoust, C., The Magma algebra system. The computational algebra group. http://magma.maths.usyd.edu.au/magma/. [ Links ]
[9] Cabanes, M., `Brauer morphism between modular Hecke algebras´, Journal of Algebra 115, (1988), 1-31. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{Cortés-MedinaValero-Elizondo07, ]]>
AUTHOR = {Adán Cortés-Medina and Luis Valero-Elizondo},
TITLE = {{A computacional verification of Alperin's weight conjecture for groups of small order and their prime fields}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2007},
volume = {41},
number = {2},
pages = {325-331}
}