## Hecke algebras for the basic characters of the unitriangular group

HTML articles powered by AMS MathViewer

- by Carlos A. M. André PDF
- Proc. Amer. Math. Soc.
**132**(2004), 987-996 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

## Additional 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