Automorphisms and the fundamental operators associated with the symmetrized tridisc

The automorphisms of the symmetrized polydisc $\mathbb G_n$ are well-known and are given in the coordinates of the polydisc in \cite{E:Z}. We find an explicit formula for the automorphisms of $\mathbb G_n$ in its own coordinates. If $\tau$ is an automorphism of $\mathbb G_n$, then $\tau(S_1,\dots,S_{n-1},P)$ is a $\Gamma_n$-contraction, where a $\Gamma_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set. Corresponding to every $\Gamma_n$-contraction $(S_1,\dots,S_{n-1},P)$, there exist $n-1$ unique operators $A_1,\dots,A_{n-1}$ such that $$S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad D_P=(I-P^*P)^{1/2}\,,$$ for $i=1,\dots, n-1$. This unique $(n-1)$-tuple $(A_1,\dots,A_{n-1})$, which is called the fundamental operator tuple or $\mathcal F_O$-tuple of $(S_1,\dots,S_{n-1},P)$ in literature, plays central role in every section of operator theory on $\Gamma_n$. We find an explicit form of the $\mathcal F_O$-tuple of $\tau (S_1,\dots,S_{n-1},P)$ when $n=3$. We show by an example that a $\Gamma_n$-contraction may not have commuting $\mathcal F_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $\Gamma_n$-contractions are unitarily equivalent.