No CrossRef data available.
Article contents
ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS
Part of:
Metric geometry
Representation theory of groups
Basic linear algebra
Special aspects of infinite or finite groups
Algebraic combinatorics
Published online by Cambridge University Press: 06 October 2023
Abstract
In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
Bourbaki, N., Lie Groups and Lie Algebras: Chapters 4–6, Elements of Mathematics (Springer-Verlag, Berlin, 2002); translated from the 1968 French original by A. Pressley.CrossRefGoogle Scholar
Chevalley, C., Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Actualités Scientifiques et Industrielles, 1226 (Hermann & Cie, Paris, 1955).Google Scholar
Curtis, C. W., Iwahori, N. and Kilmoyer, R. W., ‘Hecke algebras and characters of parabolic type of finite groups with (B,N)-pairs’, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 81–116.CrossRefGoogle Scholar
Fulton, W. and Harris, J., Representation Theory. A First Course, Graduate Texts in Mathematics, 129 (Springer-Verlag, New York, 1991).Google Scholar
Geck, M. and Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras, London Mathematical Society Monographs, New Series, 21 (Clarendon Press, Oxford–New York, 2000).Google Scholar
Hu, H., ‘Reflection representations of Coxeter groups and homology of Coxeter graphs’, Preprint, 2023, arXiv:2306.12846.Google Scholar
Kane, R., Reflection Groups and Invariant Theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5 (Springer-Verlag, New York, 2001).CrossRefGoogle Scholar
Milne, J. S., Algebraic Groups. The Theory of Group Schemes of Finite Type over a Field, Cambridge Studies in Advanced Mathematics, 170 (Cambridge University Press, Cambridge, 2017).CrossRefGoogle Scholar
Steinberg, R., Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar