The fundamental invariant of the Hecke algebra H n (q) characterizes the representations of H n (q), S n , SU q (N), and SU(N)

The irreducible representations (irreps) of the Hecke algebra H n (q) are shown to be completely characterized by the fundamental invariant of this algebra, C n . This fundamental invariant is related to the quadratic Casimir operator, C 2, of SU q (N), and reduces to the transposition class‐sum, [(2)] n , of S n when q → 1. The projection operators constructed in terms of C n for the various irreps of H n (q) are well behaved in the limit q → 1, even when approaching degenerate eigenvalues of [(2)] n . In the latter case, for which the irreps of S n are not fully characterized by the corresponding eigenvalue of the transposition class‐sum, the limiting form of the projection operator constructed in terms of C n gives rise to factors that depend on higher class‐sums of S n , which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class‐sums of S n , the coefficients constitute the corresponding row in the character table of S n . The properties of the fundamental invariant are used to formulate a simple and efficient recursive procedure for the evaluation of the traces of the Hecke algebra. The closely related quadratic Casimir operator of SU q (N) plays a similar role, providing a complete characterization of the irreps of SU q (N) and—by constructing appropriate projection operators and then taking the q → 1 limit—those of SU(N) as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.