Products of transvections in one conjugacy class in the symplectic group over \(\text{GF}(3)\)
type: Item
instance of: scholarly article
title: Products of transvections in one conjugacy class in the symplectic group over \(\text{GF}(3)\)
zbMATH Open document ID: 0809.20035
DOI: 10.1016/0024-3795(94)90181-3
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: The symplectic group \(\text{Sp} (2n,q)\), \(q\) odd, is generated by transvections. In a former paper \textit{E. W. Ellers} and \textit{J. Malzan} [Linear Algebra Appl. 146, 121-132 (1991; Zbl 0718.20022)] showed that every symplectic transformation is a product of transvections in one conjugacy class \(\Lambda\) of transvections, and solved the length problem for the groups \(\text{Sp} (2n,q)\), \(q\) odd, \(q \neq 3\), with respect to these \(\lambda\)-transvections. Now the author tackles the case \(q = 3\) which is more difficult than the case \(q \neq 3\), since there are not only \(\lambda\)-standard and \(\lambda\)-exceptional transformations, but also \(\lambda\)-sporadic ones. Moreover, there are two conjugacy classes of \(\lambda\)-standard, three conjugacy classes of \(\lambda\)-exceptional, and seven classes of \(\lambda\)-sporadic transformations. In \(\text{Sp} (2n,3)\) the \(\lambda\)- length, i.e. the minimal number of \(\lambda\)-transvections needed to express a given transformation, of a symplectic transformation \(A\) is equal to the codimension \(m\) of the space of vectors fixed under \(A\), or equal to \(m + 1\) if \(A\) is \(\lambda\)-standard. For \(\lambda\)-exceptional or \(\lambda\)-sporadic transformations the \(\lambda\)-length is \(m + 1\) or \(m + 2\).
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 614107
zbMATH Keywords: conjugacy class of transvections
zbMATH Keywords: symplectic group
zbMATH Keywords: generated by transvections
zbMATH Keywords: symplectic transformation
zbMATH Keywords: product of transvections
zbMATH Keywords: length problem
author: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90181-3
OpenAlex ID: W2010259304
cites work: Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries
cites work: Products of positive transvections in the real symplectic group
cites work: Products of \(\lambda\)-transvections in Sp(2n,q)
cites work: Isometrien von Vektorräumen. I
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description: scientific article; zbMATH DE number 614107