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

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... 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.