Hecke algebras for the basic characters of the unitriangular group
HTML articles powered by AMS MathViewer
- by Carlos A. M. André
- Proc. Amer. Math. Soc. 132 (2004), 987-996
- DOI: https://doi.org/10.1090/S0002-9939-03-07143-0
- Published electronically: July 17, 2003
- PDF | Request permission
Abstract:
Let $U_{n}(q)$ denote the unitriangular group of degree $n$ over the finite field with $q$ elements. In a previous paper we obtained a decomposition of the regular character of $U_{n}(q)$ as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character $\xi _{{\mathcal {D}}}(\varphi )$ of $U_{n}(q)$. We prove that $\xi _{ {\mathcal {D}}}(\varphi )$ is induced from a linear character of an algebra subgroup of $U_{n}(q)$, and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of $\xi _{{\mathcal {D}}}(\varphi )$ as characters induced from an algebra subgroup of $U_{n}(q)$. Finally, we identify a special irreducible constituent of $\xi _{{\mathcal {D}}}(\varphi )$, which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption $p \geq n$ where $p$ is the characteristic of the field) that gives a necessary and sufficient condition for $\xi _{{\mathcal {D}}}(\varphi )$ to have a unique irreducible constituent.References
- Carlos A. M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995), no. 1, 287–319. MR 1338979, DOI 10.1006/jabr.1995.1187
- C. A. M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130, no. 7, (2002), 1943–1954.
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- Bertram Huppert, Character theory of finite groups, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. MR 1645304, DOI 10.1515/9783110809237
- I. M. Isaacs, Characters of groups associated with finite algebras, J. Algebra 177 (1995), no. 3, 708–730. MR 1358482, DOI 10.1006/jabr.1995.1325
- I. M. Isaacs and Dikran Karagueuzian, Conjugacy in groups of upper triangular matrices, J. Algebra 202 (1998), no. 2, 704–711. MR 1617655, DOI 10.1006/jabr.1997.7311
Bibliographic Information
- Carlos A. M. André
- Affiliation: Departamento de Matemática e Centro de Estruturas Lineares e Combinatórias, Faculdade de Ciências da Universidade de Lisboa, Rua Ernesto de Vasconcelos, Edifício C1, Piso 3, 1749-016 Lisboa, Portugal
- Email: candre@fc.ul.pt
- Received by editor(s): September 26, 2002
- Received by editor(s) in revised form: December 3, 2002
- Published electronically: July 17, 2003
- Communicated by: Stephen D. Smith
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 987-996
- MSC (2000): Primary 20C15; Secondary 20G40
- DOI: https://doi.org/10.1090/S0002-9939-03-07143-0
- MathSciNet review: 2045413