Title: The failure of rational dilation on the symmetrized $n$-disk for any $n\geq 3$,
Abstract: The open and closed \textit{symmetrized polydisc} or, \textit{symmetrized
$n$-disc} for $n\geq 2$, are the following subsets of $\mathbb C^n$:
\begin{align*} \mathbb G_n &=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq
i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|< 1, i=1,\dots,n
\right \}, \Gamma_n & =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq
i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n
\right \}. \end{align*} A tuple of commuting $n$ operators
$(S_1,\dots,S_{n-1},P)$ defined on a Hilbert space $\mathcal H$ for which
$\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. In this
article, we show by a counter example that rational dilation fails on the
symmetrized $n$-disc for any $n\geq 3$. We find new characterizations for the
points in $\mathbb G_n$ and $\Gamma_n$. We also present few new
characterizations for the $\Gamma_n$-unitaries and $\Gamma_n$-isometries.