Characterizations of Symmetrized Polydisc

Let ${\mathrm{\Gamma }}_n$, $n\ge 2$, denote the symmetrized polydisc in ${\mathbb{C}}^n$, and ${\mathrm{\Gamma }}_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in ${\mathrm{\Gamma }}_n$. In particular, an element $(s_1,\dots ,s_{n-1},p)\in {\mathbb{C}}^n$ is in ${\mathrm{\Gamma }}_n$ if and only if $s_j={\beta }_j+\overline{{\beta }_{n-j}}p$, $j=1,\dots ,n-1$, for some $({\beta }_1,\dots ,{\beta }_{n-1})\in {\mathrm{\Gamma }}_{n-1}$, and $\mathrm{\mid }p\mathrm{\mid }\le 1$.