Distinguished varieties in a family of domains associated with spectral interpolation and operator theory

We find characterization for the distinguished varieties in symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on G_2$ by author Shalit. show that a variety $\Lambda$ G_n$ is part of an algebraic curve, which set-theoretic complete intersection, can be represented Taylor joint spectrum $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ $\Gamma_n=\overline{\mathbb G_n}$ spectral set called $\Gamma_n$-contraction. To every $\Gamma_n$-contraction there unique operator tuple $(F_1, , F_{n-1})$, $\mathcal F_O$-tuple ,S_{n-1},P)$, \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, ,n-1. \] produce concrete functional model pure isometric-operator tuples associated with $\Gamma_n$ application we establish $\Gamma_n$-contractions $(S_1^*, S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilations $\Lambda_{\Sigma}$ G_n$, when determined ,S_{n-1}, P)$. Further, dilation ,S_{n-1}^*,P^*)$ minimal acts unitary $P^*$. Also, interplay between G_{3}$.