On the representations of \(q\) unipotent groups over a finite field.
type: Item
instance of: scholarly article
title: On the representations of \(q\) unipotent groups over a finite field.
zbMATH Open document ID: 1253.20045
author: David Ginzburg
published in: Israel Journal of Mathematics
publication date: 2011-05-05T00:00:00Z
review text: Let \(q\) denote a power of a prime, and \(F_q\) denote the finite field with \(q\) elements. Let \(U_n(q)\) denote the finite group which consists of all upper triangular matrices of size \(n\) with ones along the diagonal. If \(U_n(q)\) has an irreducible representation of degree \(q^e\), then \(e\leq\mu(n)\), where \(\mu(n)=m(m-1)\) if \(n=2m\) and \(\mu(n)=m^2\) if \(n=2m+1\). Denote by \(N_{n,e}(q)\) the number of irreducible representations of degree \(q^e\). There are certain conjectures related to these numbers. In this paper, the author studies some conjectures related to the group \(U_n(q)\) and some of its subgroups. He shows that the numbers \(N_{n,e}(q)\) are all polynomials in \(q-1\). Concretely, in the first two sections he verifies the above conjecture for some \(q\) unipotent groups. In section 2 he proves it for the generalized Heisenberg groups, and for the maximal unipotent radicals of the classical groups \(\text{Sp}_6\) and \(\text{SO}_8\). In section 3 he verifies it also for the maximal unipotent subgroup of the exceptional group \(G_2\). In section 4 he tries to give such an interpretation for the group \(U_n\).
reviewed by: Nan Ji-Zhu
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 5884608
zbMATH Keywords: representations of unipotent groups
zbMATH Keywords: numbers of irreducible representations
zbMATH Keywords: degrees
zbMATH Keywords: finite fields
zbMATH Keywords: polynomial functions
MaRDI profile type: Publication
full work available at URL: s11856-011-0015-7
OpenAlex ID: W1976779152
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DOI: 10.1007/S11856-011-0015-7
description: scientific article; zbMATH DE number 5884608