  Symmetric homology is a natural generalization of cyclic homology, in which
symmetric groups play the role of cyclic groups. In the case of associative
algebras, the symmetric homology theory was introduced by Z. Fiedorowicz
\cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In
this paper, we show that, for algebras defined over a field of characteristic
$0$, the symmetric homology theory is naturally equivalent to the
(one-dimensional) representation homology theory introduced by the authors
(jointly with G. Khachatryan) in \cite{BKR}. Using known results on
representation homology, we compute symmetric homology explicitly for basic
algebras, such as polynomial algebras and universal enveloping algebras of (DG)
Lie algebras. As an application, we prove two conjectures of Ault and
Fiedorowicz, including the main conjecture of \cite{AF07} on topological
interpretation of symmetric homology of polynomial algebras.
