Lifting maps from the symmetrized polydisk in small dimensions

Abstract

The spectral unit ball $\Omega_n$ is the set of all $n\times n$ matrices with spectral radius less than $1$. Let $\pi(M) \in \mathbb C^n$ stand for the coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisk is $\mathbb G_n:=\pi(\Omega_n)$. When investigating Nevanlinna-Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisk (for instance to obtain continuity results for the Lempert function): if $\psi \in \mathcal O(\mathbb D, \Omega_n)$, then $\pi \circ \psi \in \mathcal O(\mathbb D, \mathbb G_n)$. Given a map $\varphi \in \mathcal O(\mathbb D, \mathbb G_n)$, we are looking for necessary and sufficient conditions for this map to "lift through given matrices", i.e. find $\psi$ as above so that $\pi \circ \psi = \varphi$ and $\psi (\alpha_j) = M_j$, $1\le j \le N$. A natural necessary condition is $\varphi(\alpha_j)=\pi(M_j)$, $1\le j \le N$. When the matrices $M_j$ are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of $\varphi$ at the points $\alpha_j$. Those conditions are necessary and sufficient for a local lift. We give a scheme which shows that the necessary conditions are also sufficient for a global lift in small dimensions (up to $5$), and a counter-example to show that the scheme fails in dimension $6$ (and above).