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Hecke algebras for the basic characters of the unitriangular group


Author: Carlos A. M. André
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
Published electronically: July 17, 2003
MathSciNet review: 2045413
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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_{{\EuScript{D}}}(\varphi)$ of $U_{n}(q)$. We prove that $\xi_{ {\EuScript{D}}}(\varphi)$ 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 $\xi_{{\EuScript{D}}}(\varphi)$ as characters induced from an algebra subgroup of $U_{n}(q)$. Finally, we identify a special irreducible constituent of $\xi_{{\EuScript{D}}}(\varphi)$, 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_{{\EuScript{D}}}(\varphi)$ to have a unique irreducible constituent.


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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

DOI: https://doi.org/10.1090/S0002-9939-03-07143-0
Keywords: Unitriangular group, irreducible character, basic character
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
Article copyright: © Copyright 2003 American Mathematical Society

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