Hostname: page-component-f554764f5-fnl2l Total loading time: 0 Render date: 2025-04-23T00:27:04.115Z Has data issue: false hasContentIssue false
  • English
  • Français

ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS

Part of: Basic linear algebra Algebraic combinatorics Special aspects of infinite or finite groups Representation theory of groups Metric geometry

Published online by Cambridge University Press:  06 October 2023

HONGSHENG HU Open the ORCID record for HONGSHENG HU [Opens in a new window]
HONGSHENG HU*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, PR China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

exterior powersgeneralised reflectionsreflection representations

MSC classification

Primary: 15A75: Exterior algebra, Grassmann algebras
Secondary: 05E10: Combinatorial aspects of representation theory 20C15: Ordinary representations and characters 20F55: Reflection and Coxeter groups 51F15: Reflection groups, reflection geometries
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 1 , August 2024 , pp. 90 - 102
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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.Google 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).CrossRefGoogle Scholar
Hu, H., ‘Reflection representations of Coxeter groups and homology of Coxeter graphs’, Preprint, 2023, arXiv:2306.12846.CrossRefGoogle Scholar
Kane, R., Reflection Groups and Invariant Theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5 (Springer-Verlag, New York, 2001).Google 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).Google Scholar
Steinberg, R., Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar