  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, ${\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition
class-sum, $[(2)]_n$, of $S_n$ when $q\rightarrow 1$. The projection operators
constructed in terms of $C_n$ for the various irreps of $H_n(q)$ are
well-behaved in the limit $q\rightarrow 1$, 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
$q\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic
Casimir operator of the latter does not suffice to specify its irreps.
