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.

When? | 19.12.2023 17:15 |
---|---|

Where? | PER 08 auditoire 2.52 Chemin du Musée 3, 1700 Fribourg |

speaker | Prof. Frank Kutzschebauch, Uni Bern |

Contact | Département de mathématiques isabella.schmutz@unifr.ch |