  We study a finite-dimensional quotient of the Hecke algebra of type $H_n$ for
general $n$, using a calculus of diagrams. This provides a basis of monomials
in a certain set of generators. Using this, we prove a conjecture of C.K. Fan
about the semisimplicity of the quotient algebra. We also discuss the cellular
structure of the algebra, with certain restrictions on the ground ring.
