  The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed
class-sum of transpositions of the symmetric group $S_{n}$. Irreducible
representations of $H_{n}(q)$, for generic $q$, are shown to be completely
characterized by the corresponding eigenvalues of $C_{n}$ alone. For $S_{n}$
more and more invariants are necessary as $n$ inereases. It is pointed out that
the $q$-deformed classical quadratic Casimir of $SU(N)$ plays an analogous
role. It is indicated why and how this should be a general phenomenon
associated with $q$-deformation of classical algebras. Apart from this
remarkable conceptual aspect $C_{n}$ can provide powerful and elegant
techniques for computations. This is illustrated by using the sequence $C_{2}$,
$C_{3}, \cdots,\; C_{n}$ to compute the characters of $H_{n}(q)$.
