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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hecke algebras for the basic characters of the unitriangular group
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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.
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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
  • 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