New expressions for the invariant operators of the unitary groups

The invariant operators (or Casimir operators) for the unitary groups $U(n)$ and $SU(n)$ are considered. The eigenvalues of these operators for an arbitrary irreducible representation are expanded with respect to standard power sums $S_k$ defined by Eq. (2.8). For the coefficients $\beta_p(\nu)$ of this expansion the expressions (3.9), (3,17), and (3.18) are obtained; they holed for arbitrary rank $n-1$ of the group and arbitrary order $p$ of the invariant operator. These expressions considerably simplify the calculation of the eigenvalues of the invariant operators (especially for large $p$), which is demonstrated by a number of examples. The connection between the operators (2.1) and (5.3), which correspond to different ways of contracting indices, is found.