Jerry Malzan

Products of λ-transvections in Sp(2n,q)

Abstract Every transvection A in the symplectic group Sp(2 n , q ), q a power of an odd prime, is a product of transvections T i , where all T i are in the same conjugacy class C of transvections. For each transformation A we determine the length of A , i.e., we find the least number of transvections in C required to express A .

Products of positive transvections in the real symplectic group

Each transformation Ain the real symplectic group is a product of positive transvections. For any Awe determine its length, i.e. the minimal number of factors needed in such a product.

Orbits of Nilpotent Matrices

Abstract The action under conjugation of invertible lower triangular n × n matrices (over an infinite field) on lower triangular nilpotent matrices, N n , divides N n into orbits. We show that for n ⩾6 the number of orbits is infinite.

Perturbation of eigenvalues in U(n) by pseudo-reflections

A unitary pseudo-reflection is a matrix where a is a unit vector in Cn . If is a unitary matrix we study the variation of eigenvalues of ARθ when θ increases from 0 to 2π. Some of these eigenvalues may remain fixed while the others move monotonicaliy and counter-clockwise along the unit circle. As a byproduct, we obtain an explicit formula for the determinant of the matrix whose (i,j)th entry is .

Products of reflections in the quaternionic unitary group

Let Sp(n) denote the unitary group of a positive definite hermitian form on an n-dimensional right quaternionic vector space. Thus Sp(n) is the compact real form of the complex symplectic group Sp(2n, C). For each A ϵ Sp(n), n ⩾ 2, we determine the smallest number m such that A can be written as a product of m reflections in Sp(n). In particular, we show that if n ⩾ 4 then every A ϵ Sp(n) is a product of n, n + 1, or n + 2 reflections. In the projective group PSp(n), n > 4, every element is a product of n reflections.

Transvections as generators of the special linear group over the quaternions

Let V be a finite-dimensional right vector space over the quaternions H. Each transformation M in the special linear group of V is a product of transvections Ti, i.e. M = T1…Tt. The smallest t is called the length of M, t = l(M). We show that l(M) = dim B(M) ≥3, M≠3J1(λ) where λ ϵ C\R, and Mμ is neither simple nor the identity for any μ ϵ R. In any case l(M) = dim B(M) + i, where 0≤i≤3.