  Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ and for an integer $d\geq 1$,
let $S_d$ denote the symmetric group, consisting of of all permutations of the
set $\{1,\cdots, d\}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is
symmetric if $f(z_1,\cdots, z_d)=f(z_{\sigma(1)},\cdots, z_{\sigma (d)})$ for
all $\sigma \in S_d$ and all $(z_1,\cdots, z_d)\in {\mathbb{D}}^d$. The
polydisc algebra $A({\mathbb{D}}^d)$ is the Banach algebra of all holomorphic
functions $f$ on the polydisc ${\mathbb{D}}^d$ that can be continuously
extended to the closure of the polydisc in ${\mathbb{C}}^d$, with pointwise
operations and the supremum norm (given by $\|f\|_\infty:=\sup_{\mathbf{z} \in
{\mathbb{D}}^d} |f(\mathbf{z})|$). Let $A_{\textrm{sym}}({\mathbb{D}}^d)$ be
the Banach subalgebra of $A({\mathbb{D}}^d)$ consisting of all symmetric
functions in the polydisc algebra. Algebraic-analytic properties of
$A_{\textrm{sym}}({\mathbb{D}}^d)$ are investigated. In particular, the
following results are shown: the corona theorem, description of the maximal
ideal space and its contractibility, Hermiteness, projective-freeness, and
non-coherence.
