J ul 2 00 1 The Structure of The Group of Polynomial Matrices Unitary In The Indefinite Metric of Index 1

We consider the group M of all polynomial ν × ν matrices U(ω) = ∑κι=0 Uι ω, κ = 0, 1, 2, ..., that satisfy equation U(ω)·D·U(ω)∗ = D with diagonal ν×ν matrixD = diag{−1, 1, 1, ...1}. Here ν ≥ 2, U(ω) = ∑κι=0 U ι ω and U ι is the Hermitean conjugate of Uι. We show that the subgroup M0 of those U(ω) ∈ M, that are normalized by the condition U(0) = I, is the free product of certain groups Mz where z is a ν-vector drawn from the set Ξ def = {z = column{ζ1, ζ2, ...ζν} | ζ1 = 1, zDz = 0}. Matrices in each Mz are explicitly and uniquely parametrized in the paper. Thus, every ν × ν matrix U = U(ω) ∈ M0 can be represented in the form U = Gz1 · Gz2 · ... · Gzη with ν × ν polynomial matrix multiples Gzι ∈ Mzι , zι ∈ Ξ, for ι = 1, ...η, so that zι 6= zι+1 for ι = 1, ...η−1 and this representation is unique. The uniqueness includes the number of multiples η where η = 0, 1, 2, ..., their particular sequence Gz1 , Gz2 , ...Gzη , and the multiples themselves with their respective parametrizations; all these items can be defined in only one way once the U is given.