  Third order three-dimensional symmetric and traceless tensors play an
important role in physics and tensor representation theory. A minimal integrity
basis of a third order three-dimensional symmetric and traceless tensor has
four invariants with degrees two, four, six and ten respectively. In this
paper, we show that any minimal integrity basis of a third order
three-dimensional symmetric and traceless tensor is also an irreducible
function basis of that tensor, and there is no polynomial syzygy relation among
the four invariants of that basis, i.e., these four invariants are
algebraically independent.
