We present new efficient data structures for elements of Coxeter groups of type Am and their associated Iwahori–Hecke algebras H(Am). Usually, elements of H(Am) 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(Am) 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(Am).