Untriangular factorization of holomorpic symplectic matrices

TL;DR

This paper proves that every holomorphic symplectic matrix can be factorized into a product of holomorphic unitriangular matrices, providing estimates on the number of factors and offering a simpler proof of existing factorizations.

Contribution

It introduces a new factorization method for holomorphic symplectic matrices and estimates the minimal number of factors needed, solving an open problem.

Findings

Every holomorphic symplectic matrix can be factorized into holomorphic unitriangular matrices.

Holomorphic unitriangular matrices are products of at most 7 such matrices with respect to the standard form.

Provides a simpler, more elegant proof of existing matrix factorizations.

Abstract

We prove that every holomorphic symplectic matrix can be factorized as a product of holomorphic unitriangular matrices with respect to the symplectic form $ \left[\begin{array}{ccc} 0 & L_n \\ -L_n & 0\end{array}\right]$ where $L$ is the $n \times n$ matrix with $1$ along the skew-diagonal. Also we prove that holomorphic unitriangular matrices with respect to this symplectic form are products of not more than $7$ holomorphic unitriangular matrices with respect to the standard symplectic form $\left[\begin{array}{ccc} 0 & I_n \\ -I_n & 0\end{array}\right]$, thus solving an open problem posed in \cite{HKS}. Combining these two results allows for estimates of the optimal number of factors in the factorization by holomorphic unitriangular matrices with respect to the standard symplectic form. The existence of that factorization was obtained earlier by Ivarsson-Kutzschebauch and Schott, however without any estimates. Another byproduct of our results is a new, much less technical and more elegant proof of this factorization.