Characterizations of the symmetrized polydisc via another family of domains

We find new characterizations for the points in the \textit{symmetrized polydisc} ${\mathbb{G}}_n$, a family of domains associated with the spectral interpolation, defined by $${\mathbb{G}}_n:=\{(\sum _{1\le i\le n}{z}_i,\sum _{1\le i<j\le n}{z}_i{z}_j\dots ,\prod _{i=1}^n{z}_i):\mathrm{\mid }{z}_i\mathrm{\mid }<1,i=1,\dots ,n\}\mathrm{.}$$ We introduce a new family of domains which we call $\mathit{\text{the}}\mathit{\text{extended}}\mathit{\text{symmetrized}}\mathit{\text{polydisc}}$ ${\tilde{\mathbb{G}}}_n$, and define in the following way: \begin{align*} \widetilde{\mathbb G}n := \Bigg{ (y_1,\dots,y{n-1}, q)\in \mathbb C^n :; q \in \mathbb D, ; y_j = \beta_j + \bar{\beta}{n-j} q, ; \beta_j \in \mathbb C &$\text{ and }$\ |\beta_j|+ |\beta{n-j}| < {n \choose j} &$\text{ for }$ j=1,\dots, n-1 \Bigg}. \end{align*} We show that ${\mathbb{G}}_n={\tilde{\mathbb{G}}}_n$ for $n=1,2$ and that ${\mathbb{G}}_n⊊{\tilde{\mathbb{G}}}_n$ for $n\ge 3$. We first obtain a variety of characterizations for the points in $\widetilde{\mathbb G}_n$ and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in ${\mathbb{G}}_n$. Also we obtain similar characterizations for the points in $\Gamma_n \setminus {\mathbb G}_n$,where$\Gamma_n =\overline{{\mathbb G}_n}$\mathrm{.}Asetof$n-1$ fractional linear transformations play central role in the entire program. We also show that for $n\ge 2$, ${\tilde{\mathbb{G}}}_n$ is non-convex but polynomially convex and is starlike about the origin but not circled.