Products of transvections in one conjugacy class in the symplectic group over \(\text{GF}(3)\) (Q1329921)

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 \(m + 2\).