  We prove some estimates for elementary symmetric polynomials on $\mathbb
D^n.$ We show that these estimates are sharp which allow us to study the
properties of closed symmetrized polydisc $\Gamma_n.$ Furthermore, we show the
existence and uniqueness of solutions to the operator equations
$$S_i-S_{n-i}^*S_n=D_{S_n}X_iD_{S_n}~~{\rm{and}}~~S_{n-i}-S_{i}^*S_n=D_{S_n}X_{n-i}D_{S_n},$$
where $X_i,X_{n-i}\in \mathcal B(\mathcal D_{S_n}), ~{\rm{for ~all~}}
i=1,\ldots,(n-1),$ with numerical radius not greater than $1,$ for a
$\Gamma_n$-contraction $(S_1,\ldots, S_n).$ We construct a conditional dilation
of various classes of $\Gamma_n$-contractions. Various properties of a
$\Gamma_n$-contraction and its explicit dilation allow us to construct a
concrete functional model for a $\Gamma_n$-contraction. We describe the
structure and additional characterization of $\Gamma_n$-unitaries and
$\Gamma_n$-isometries in detail.
