A large deviation principle is proved for a family of measures
{
L
n
:
n
=
1
,
2
,
…
}
\left \{ {{\mathbb {L}_n}:n = 1,2, \ldots } \right \}
derived from the multiplicities occurring in the decomposition into irreducible components of
n
n
-fold tensor products of representations of arbitrary compact semisimple Lie groups.