Characterizations of Symmetrized Polydisc

Let $\Gamma_n$, $n \geq 2$, denote the symmetrized polydisc in $\mathbb{C}^n$, and $\Gamma_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in $\Gamma_n$. In particular, an element $(s_1, \ldots, s_{n-1}, p) \in \mathbb{C}^n$ is in $\Gamma_n$ if and only if $s_j = \beta_j + \overline{\beta_{n-j}} p$, $j = 1, \ldots, n-1$, for some $(\beta_1, \ldots, \beta_{n-1}) \in \Gamma_{n-1}$, and $|p| \leq 1$.