Products of transvections in one conjugacy class in the symplectic group over GF(3)

Abstract

The transvections in any one conjugacy class Λ generate the symplectic group Sp(2 n, 3) over GF(3). The elements in Λ are called λ-transvections. There is a partition of the symplectic group into λ-standard, λ-exceptional, and λ-sporadic transformations. There are two conjugacy classes of λ-standard, three conjugacy classes of λ-exceptional, and seven classes of λ-sporadic transformations. We determine the λ-length of each symplectic transformation A, i.e., we determine l λ ( A) = t such that there are λ-transvections T i with A = T 1 ⋯ T t , but A is not a product of fewer than t λ-transvections.