Every complex symplectic matrix in Sp2n
pCq can be factorized as a
product of the following types of unipotent matrices (in interchanging
order).
‚ (i): ˆ
I B
0 I
˙
, upper triangular with symmetric B “ BT
.
‚ (ii): ˆ
I 0
C I˙
, lower triangular with symmetric C “ C
T
.
The optimal number TpCq of such factors that any matrix in Sp2n
pCq
can be factored into a product of T factors has recently been established
to be 5 by Jin, P. Lin, Z. and Xiao, B.
If the matrices depend continuously or holomorphically on a parameter, equivalently their entries are continuous functions on a topological
space or holomorphic functions on a Stein space X, it is by no means
clear that such a factorization by continuous/holomorphic unipotent
matrices exists. A necessary condition for the existence is the map
X Ñ Sp2n
pCq to be null-homotopic. This problem of existence of a
factorization is known as the symplectic Vaserstein problem or GromovVaserstein problem. In this talk we report on the results of the speaker
and his collaborators B. Ivarsson, E. Low and of his Ph.D. student J.
Schott on the complete solution of this problem, establishing uniform
bounds Tpd, nq for the number of factors depending on the dimension
of the space d and the size n of the matrices. It seems difficult to establish the optimal bounds. However we obtain results for the numbers
Tp1, nq, Tp2, nq for all sizes of matrices in joint work with our Ph.D.
students G. Huang and J. Schott. Finally we give an application to
the problem of writing holomorphic symplectic matrices as product of
exponentials.