A Lower Bound for the Cardinality of Function Basis of Tensor Invariants

In this article, we give a proof for that the cardinality of a function basis of the invariants for a finite dimensional real vector space by a compact group is lower bounded by the intuitive difference of the dimensions of the vector space and the group. An application is given to the space of third order three dimensional symmetric and traceless tensors, showing that each minimal integrity basis is an irreducible function basis, which solves a problem in applied mechanics.