Let
(
R
,
m
)
(R,m)
be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free
R
R
-module
A
A
, write
A
n
A_{n}
for the
n
n
th symmetric power of
A
A
, mod torsion. We study the modules
A
n
A_{n}
,
n
≥
1
n \geq 1
, when
A
A
is complete (i.e., integrally closed). In particular, we show that
B
⋅
A
=
A
2
B\cdot A = A_{2}
, for any minimal reduction
B
⊆
A
B \subseteq A
and that the ring
⊕
n
≥
1
A
n
\oplus _{n \geq 1} A_{n}
is Cohen-Macaulay.