Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T12:45:20.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructing representations of Hecke algebras for complex reflection groups

Part of: Computational aspects of associative rings Representation theory of groups

Published online by Cambridge University Press:  01 August 2010

Gunter Malle  and
Jean Michel
Gunter Malle
Affiliation:
FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (email: [email protected])
Jean Michel
Affiliation:
CNRS, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France (email: [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 investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of aW-graph to the situation of complex reflection groups. We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras.

MSC classification

Secondary: 16Z05: Computational aspects of associative rings 20C08: Hecke algebras and their representations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 426 - 450
Copyright
Copyright © London Mathematical Society 2010

References

[1] Ariki, S., ‘Representation theory of a Hecke algebra of G(r,p,n)’, J. Algebra 177 (1995) 164–185.CrossRefGoogle Scholar
[2] Ariki, S. and Koike, K., ‘A Hecke algebra of (Z/rZ)≀𝔖n and construction of its irreducible representations’, Adv. Math. 106 (1994) 216–243.CrossRefGoogle Scholar
[3] Bessis, D., ‘Zariski theorems and diagrams for braid groups’, Invent. Math 145 (2001) 487–507.CrossRefGoogle Scholar
[4] Bessis, D., ‘Finite complex reflection arrangements are K(π,1)’, 2008, arXivmath/0610777.Google Scholar
[5] Bessis, D. and Michel, J., ‘Explicit presentations for exceptional braid groups’, Experiment. Math. 13 (2004) 257–266.CrossRefGoogle Scholar
[6] Bremke, K. and Malle, G., ‘Reduced words and a length function for G(e,1,n)’, Indag. Math. 8 (1997) 453–469.CrossRefGoogle Scholar
[7] Broué, M. and Malle, G., ‘Zyklotomische Heckealgebren’, Astérisque 212 (1993) 119–189.Google Scholar
[8] Broué, M., Malle, G. and Michel, J., ‘Towards spetses I’, Transform. Groups 4 (1999) 157–218.CrossRefGoogle Scholar
[9] Broué, M., Malle, G. and Rouquier, R., ‘Complex reflection groups, braid groups, Hecke algebras’, J. reine angew. Math. 500 (1998) 127–190.Google Scholar
[10] Dipper, R., James, G. and Mathas, A., ‘Cyclotomic q-Schur algebras’, Math. Z. 229 (1998) 385–416.CrossRefGoogle Scholar
[11] Etingof, P. and Rains, E., ‘Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups’, J. Algebra 299 (2006) 570–588.CrossRefGoogle Scholar
[12] Gyoja, A., ‘On the existence of a W-graph for an irreducible representation of a Coxeter group’, J. Algebra 86 (1984) 422–438.CrossRefGoogle Scholar
[13] Malle, G., ‘Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux’, Finite reductive groups: related structures and representations, Progress in Mathematics 141 (Birkhäuser, Basel, 1997) 311–332.CrossRefGoogle Scholar
[14] Malle, G., ‘On the rationality and fake degrees of characters of cyclotomic algebras’, J. Math. Sci. Univ. Tokyo 6 (1999) 647–677.Google Scholar
[15] Malle, G., ‘On the generic degrees of cyclotomic algebras’, Represent. Theory 4 (2000) 342–369.CrossRefGoogle Scholar
[16] Malle, G. and Mathas, A., ‘Symmetric cyclotomic Hecke algebras’, J. Algebra 205 (1998) 275–293.CrossRefGoogle Scholar
[17] Michel, J., ‘The GAP-part of the Chevie system’, GAP 3-package available for download from http://people.math.jussieu.fr/∼jmichel/chevie/chevie.html.Google Scholar
[18] Müller, J., ‘On exceptional cyclotomic algebras’, Private communication, June 2004.Google Scholar