  We present new efficient data structures for elements of Coxeter groups of
type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually,
elements of $H(A_m)$ are represented as simple coefficient list of length $M =
(m+1)!$ with respect to the standard basis, indexed by the elements of the
Coxeter group. In the new data structure, elements of $H(A_m)$ are represented
as nested coefficient lists. While the cost of addition is the same in both
data structures, the new data structure leads to a huge improvement in the cost
of multiplication in~$H(A_m)$.
