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

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- 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
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## 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

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## 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