A family of domains associated with $\mu$-synthesis

We introduce a family of domains — which we call the ${\mu }_{1,n}$-quotients — associated with an aspect of $\mu$-synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the ${\mu }_{1,n}$-quotient and its associated unit “${\mu }_E$-ball”. Here, ${\mu }_E$ is the structured singular value for the case $E = {[w]\oplus(z I_{n-1})\in $\mathbb{C}$^{n\times n}: z,w\in $\mathbb{C}$}$, n = 2, 3, 4,… Specifically: we show that, for such an $E$, the Nevanlinna-Pick interpolation problem with matricial data in a unit “${\mu }_E$-ball”, and in general position in a precise sense, is equivalent to a Nevanlinna-Pick interpolation problem for the associated ${\mu }_{1,n}$-quotient. Along the way, we present some characterizations for the ${\mu }_{1,n}$-quotients.