  In this thesis, I present an associative diagram algebra that is a faithful
representation of a particular Temperley--Lieb algebra of type affine $C$,
which has a basis indexed by the fully commutative elements of the Coxeter
group of the same type. The Coxeter group of type affine $C$ contains an
infinite number of fully commutative elements, and so the corresponding
Temperley--Lieb algebra is of infinite rank. With the exception of type affine
$A$, all other generalized Temperley--Lieb algebras with known diagrammatic
representations are of finite rank. In the finite rank case, counting arguments
are employed to prove faithfulness, but these techniques are not available in
the affine $C$ case. To prove faithfulness, I classify the fully commutative
elements in Coxeter groups of types $B$ and affine $C$ that are irreducible
under weak star reductions. The classification of these irreducible elements
provides the groundwork for inductive arguments that are used to prove
faithfulness. The classification of the weak star irreducible elements of type
$B$ also verifies C.K. Fan's unproved claim about about the set of fully
commutative elements in a Coxeter group of type $B$ having no generator
appearing in the left or right descent set that can be left or right cancelled,
respectively. The results of this thesis will be used to construct a trace on
the Hecke algebra of type affine $C$, which will then be used to compute
leading coefficients of certain Kazhdan--Lusztig polynomials in a non-recursive
way.
